Adam came about 6000 years ago (Saudi Salafi Mufti)
According to the largest Salafi Fatwa website, IslamQA (http://islamqa.info/en/20907 ), which provides "Authentic Prophetic Traditions," there are specific gaps between different prophets from Muhammad to Adam. By combining these gaps, it is estimated that Adam came about 6000 years ago, as per Islamic belief.
Here are the calculated gaps:
The gap between Jesus and Muhammad: 600 years, as mentioned in Sahih Bukhari. The Bible also confirms the same as according to the bible Jesus came about 2000 years ago, and Prophet Muhammad came about 1400 years ago.
The gap between Adam and Noah: 10 centuries, i.e., 1000 years, supported by an authentic tradition in Sahih Ibn Habban. It is the same in the Bible too, which states it was 1056 years.
The gap between Abraham and Moses: The Bible states a gap of 245 years (6 Generations: Abraham-Jacob-Levi-Kohath-Amran(Imran)- Moses), and Islamic sources such as Ibn Kathir (link) and Ibn Jarir Tabari (link) record exactly the same genealogy from Abraham to Moses as mentioned in the Bible.
By adding up these gaps, it is estimated that even within Islamic teachings, Adam came into existence around 6000 years ago, with a variation of +/-10%.
Al-Tabari, the age of the universe is 6000 years
Al-Tabari is the most important Muslim historian. He recorded the following Traditions regarding the creation and age of the universe:
Hitory of Tabari, Vol. 1, pp. 172-173 (Link):
According to Ibn Humayd -Yahya b. Wadih -Yahya b. Ya'qub -Hammad -Sa'id b. Jubayr -Ibn 'Abbas: This world is one of the weeks of the other world-seven thousand years. Six thousand two hundred years have already passed. (The world) will surely experience hundreds of years, during which there will be no believer in the oneness of God there.
Others said that the total extent of time is six thousand years.
Those who said this
According to Abu Hisham -Mu'awiyah b. Hisham -Sufyan -al-A'mash -Abu Salih -Kab : This world is six thousand years.
According to Muhammad b. Sahl b. 'Askar -Ismail b. 'Abd al-Karim -'Abd al-Samad b. Ma'qil i -Wahb : Five thousand six hundred years of this world have elapsed. I do not know which kings and prophets lived in every period (zaman) of those years. I asked Wahb b. Munabbih: How long is (the total duration of) this world? He replied: Six thousand years.
History of Tabari, Vol. 1, pp. 183-184 (Link):
All these facts taken together make it clear that of the two statements I have mentioned concerning the total extent of time, the one from Ibn 'Abbas, and the other from Kab, the one more likely to be correct in accordance with the information coming from the Messenger of God is that of Ibn 'Abbas transmitted here by us on his authority: The world is one of the weeks of the other world-seven thousand years.'
Consequently, because this is so and the report on the authority of the Messenger of God is sound-namely, that he reported that what remained of the time of this world during his life was half a day, or five hundred years, since five hundred years are half a day of the days, of which one is a thousand years-the conclusion is that the time of this world that had elapsed to the moment of the Prophet's statement corresponds to what we have transmitted on the authority of Abu Tha'labah al-Khushani from the Prophet, and is 6,500 years or approximately 6,500 years. God knows best!
Our statement about the duration of the periods (azman) of this world from its very beginning to its very end is the most firmly established of all the statements we have, on account of the testimony to its soundness as explained by us. Information has (also) been transmitted on the authority of the Messenger of God to prove the soundness of the statement that all of this world is six thousand years. If its chain of transmitters were sound, we would have to go no further. It is Muhammad b. Sinan al-Qazzaz'43 -'Abd al-Samad b. 'Abd al-Warithf -Zabban -'Asim' -Abu Salih-Abu Hurayrah: The Messenger of God said: AI-hugb is eighty years. The day of them is one-sixth of this world."" According to this tradition, it is clear that all of this world is six thousand years. That is because, if one day of the other world equals one thousand years and a single such day is one-sixth of this world, the conclusion would be that the total is six of the days of the other world, and that is six thousand years.
Shia Islam also claims Adam (as) came about 6000 to 7000 years ago
Please see the Tafsir al-Mizan by Shia Alim Tabatabai (link).
- Islam claims Adam (as) came about 6000 years ago
- Nevertheless, this universe is billions of years old.
- And science has already found fossils of modern humans, which are over 300,000 years old.
Side Note: What does "Qaran قرن" means?
Modern Islamic apologists claim that the difference between Noah and Abraham was 10 "Qaran قرن", which in Arabic means (1) Century and also (2) Generation. They calculate that, considering Noah's age of more than 950 years, 10 Qaran would equate to about 10,000 years.
However, this argument faces several challenges:
- All dictionaries (link) unanimously define Qaran as Century, with "generation" being a less common meaning, if used at all.
- The choice of Noah's age as the standard for a generation is arbitrary. Why not consider Abraham's age, who lived for 175 years?
- Muhammad was addressing people of his era, for whom the average generation span was not more than 100 years. The Quran and Hadith were revealed in simple language to suit the understanding of that time.
- Even if we interpret "Qaran" as "generation" and apply it only to Noah, it still does not reconcile with scientific evidence of human fossils dating back much older, around 300,000 years.
It appears that the majority of Salaf (early) Islamic scholars held the view that 'Qaran' means 'century,' and the current trend of interpreting it as 'generation' seems to be a recent attempt by Islamic apologists to avoid contradictions between science and religion.
A tradition from Tabqat Ibn Sa'ad supports this perspective, stating that there were ten Qarns between Adam and Noah, where each Qarn was a span of one hundred years. Similarly, there were ten Qarns between Noah and Abraham, each spanning one hundred years.
Tabqat Ibn Sa'ad, Account of Generations and period between Adam and Muhammad (online Link):
He (Ibn Sa`d) said: Muhammad Ibn `Umar Ibn Waqid al-Aslami informed us on the authority of more than one scholar; they said: There were ten qarns between Adam and Nuh and every qarn is a span of one hundred years. There were ten qarns between Nuh and Ibrahim and a qarn was of the span of a century. Between Ibrahim and Musa Ibn `Imrán ,there was a span of ten qarns and a qarn was equal to one hundred years.
According to Islamic traditions, Adam came to the earth about 6,000 years ago. But science refuted this Islamic story, as it found thousands of human fossils which were way older than 6,000 years and the oldest human fossils are 300,000 years old.
Thus, modern Muslims were compelled to deny their own traditions and come up with the new claim that Adam came at least 300,000 years (or even earlier).
But again, Science refuted this Islamic claim, while the "Stone Age" ended only 10,000 years ago, and the first time "Cultivation of Crops" also started only 10,000 years ago after the end of the Ice Age.
While according to the Islamic story, Adam was able to cultivate the crops and his son Qabeel offered the sacrifice of the grains of the crop, while another son Habeel sacrificed a sheep for Allah.
Another problem occurred with the Arch of Noah. According to Authentic Islamic Tradition, Noah came 10 Centuries (1,000 years) after Adam. But then how was Noah able to build that grand Naval Ship with "Stone Tools"?
- The "Stone Age" means humans were able to make primitive tools out of stones or bones only.
- While the "Bronze Age" started 5,500 years ago only. There are no bronze tools which are older than 5,500 years.
- And the "Iron Age" started 3,500 years ago only. There are no iron tools present which are older than 3,500 years.
- And the first Naval Ship is only 5,000 years old according to scientific research.
- The "Ice Age" also ended 10,000 years ago, but Islam has absolutely no "clue" about the ice age. Also, the bible has no clue about the ice age.
- Again Quran/Bible have zero clue about the dinosaur.
- Scientists have found thousands of human fossils and stone tools which are much older than 6,000 years (and going back up to 300,000 years). Link.
- And scientists also found cave paintings by human beings which are much more than 50,000 years old. Link.
In light of these scientific findings, the newer claims made by modern Muslims regarding Adam and Noah are once again refuted.
A reply by a Shia Scholar
There is no inconsistence between these two. If we have heard that Prophet Adam has been created six to seven thousand years ago, this might refer to a different and new generation of human beings which started with Prophet Adam (A.S.) and might have been preceded by numbers of generations of human beings who have gone extinct. There is a lot of evidence in hand that suggests a whole different generation of humans unlike Prophet Adam existed before him.
This is also in accord with the teachings of the Quran and Islamic Hadith. 
Allamah Tabatabai (R.A.) comments under the first ayah of surah Nisa’: “The surface meaning of the verse suggests that the term “ نفس واحدة ” is referring to Adam and “ زوجها ” to Eve, who are the mother and father of mankind. We also are from this generation  of humans and judging by the Quran, mankind’s species goes back to them. The noble verse implies that the existing generation of human beings goes back to Adam and Eve and that nobody else has played any part in producing this generation.
In the history of the Jews we read that the human species we know of doesn’t surpass a history of approximately seven thousand years which seems to be reasonable. Therefore, logic suggests the same as what history dictates. But geological experts believe the history of human beings to be millions of years.
This is their point of view, but the proof they present isn’t convincing. We don’t have a reason that proves that these fossils belong to the ancestors of this chain of human beings we witness today. We also have no reason to believe that these fossils don’t belong to a generation or kind of beings who lived before Prophet Adam (The Father of Mankind) and were instinct way before our ancestor and were never connected to the chain of human beings we are from.
The Holy Quran has not explicitly explained the emergence of mankind on the planet earth, in terms of whether we were the only generation of mankind or if there were other generations before us making us the last generation.
From paying attention to some verses of the Quran, we can slightly deduce that before the creation of Prophet Adam (The Father of Mankind) and his generation there were other human beings who lived here, where He says: “…And when your Lord said to the angels: I am going to place in the earth a khalif (representative), they asked: Will you place in it someone who will make mischief in it and shed blood”. 
Allamah Tabatabai says: “We can slightly deduce from ayah 30 of surah Al-Baqara that before the creation of Prophet Adam (A.S.) there was a different generation of human beings who caused the angels to ask: Will you place in it someone who will make mischief in it and shed blood .” 
This ayah implies that another generation of human beings was created before the generation of human beings we know and are part of what was created, which the angels were initially familiar with.
In some traditions coming from the imams (A.S.), we witness proof of the existence of eras of human beings prior to the current era.
Sheikh Saduq (A.S.) has narrated a hadith in which the imam tells the narrator: “You might believe that God Almighty has not created humans other than you. Well this is false because he has created millions of humans and you from the last generation of them.” 
In Khisal, Sheikh Saduq also narrates that Imam Baqir (A.S.) has stated: “From the time He created the earth, God Almighty has created within it, seven worlds (and made all of them instinct), none of which were of the generation of of Prophet Adam. God created all of them from the crust of this earth and created generation after generation with corresponding worlds for all generations till finally Prophet Adam was created and reproduced his descendants solely from him.  , 
On this account, if we conclude that the line of history for mankind extends to a million years based on fossils found perhaps in caves, they don’t belong to the current generation of human beings, because the story of Prophet Adam appears to be about 7000 years old. 
For more information on this topic, please refer to:
Of course, there is a contradiction between Science and Islam.
There is no proof of any human being before Adam is present in the Quran. 'At maximum, this Shia Alim tried to deduce human beings before Adam in Quran, using his "CONJECTURE", which is itself Haram in Shia Islam. The Quran claims at several places that Allah made the Quran clear and easy to understand. So, the question is, is this really a clear verse about humans before Adam?
And then he presented two Ahadith, which again nowhere stating explicitly that there were humans on this earth before Adam.
The first hadith is:
Sheikh Saduq (A.S.) has narrated a hadith in which the imam tells the narrator: “You might believe that God Almighty has not created humans other than you. Well this is false because he has created millions of humans and you from the last generation of them.”
Nowhere Shia Imam is telling about the creation of humans before Adam in this tradition. Instead of Adam, Shia Imam is telling of the creation of humans before the man whom he was talking with.
And the 2nd Hadith is:
In Khisal, Sheikh Saduq also narrates that Imam Baqir (A.S.) has stated: “From the time He created the earth, God Almighty has created within it, seven worlds (and made all of them instinct), none of which were of the generation of of Prophet Adam. God created all of them from the crust of this earth and created generation after generation with corresponding worlds for all generations till finally Prophet Adam was created and reproduced his descendants solely from him
Again, Imam is not talking about humans on this EARTH, but he is talking about humans on other earths.
The Quran claimed there are 7 heavens and 7 earths. And people have always been asking where those remaining 6 earths are, and if there are people living on those 6 earths etc. Upon that Shia Imam is claiming in this tradition that Allah created humans, who are living upon those 6 earths too.
Thus, Adam and his descendants are alone on this earth according to this hadith too.
Please also consider the fact that human beings lived for 290,000 years in the stone age (i.e. the oldest human fossils are 300,000 years old, while the stone age ended only 10.000 years ago).
How is it possible?
If Allah was sending prophets to the so-called generation before Adam, how could then they stay in the stone age for such a long period of time?
Also, human beings were hunter-gatherers till 10.000 years ago (i.e. no cultivation of crops etc.).
There are thousands of Cave-Paintings present, which are about about 50,000 years old, but none of them shows anyone praying to Allah or any mosques etc. Why?
If Adam came only 6000 years ago, then we should have a lot of archaeological evidence from 6000 to 4000 years ago in the whole world, but nowhere any monolithic God or his prophets or mosques could be seen in cave paintings, or upon skin paintings, or in stone writings except for the small area where Jews lived. Why?