Erdogan denied ever saying that and told reporters in Istanbul before Biden's apology that he "will be history for me if he has indeed used such expressions."
Biden spoke with Erdogan by phone on Saturday, the White House said.
"The vice president apologised for any implication that Turkey or other allies and partners in the region had intentionally supplied or facilitated the growth of ISIL or other violent extremists in Syria," the White House said, referring to an acronym for the Islamic State group.
Responding to questions following his speech at the Harvard Kennedy School on Thursday, Biden described Erdogan as "an old friend." Biden added: "He (Erdogan) said: 'You were right. We let too many people through.' Now they're trying to seal their border."
Erdogan said: "I have never said to him that we had made a mistake, never. If he did say this at Harvard then he has to apologize to us."
He said Turkey had prevented 6,000 suspected jihadis from entering the country and deported another 1,000.
This week Turkey's parliament approved a motion giving the government powers for military operations across the border in Syria and Iraq and for foreign troops to use Turkey's territory.
On Friday, the two men also held a telephone discussion of ways their two countries "can work together to degrade and destroy and restore security and stability to the region," according to the White House.
"The Turks, who are great friends I have a great relationship with Erdogan, whom I spend a lot of time with - the Saudis, the Emiratis, etc. What were they doing? They were so determined to take down Assad and essentially have a proxy Sunni-Shia war," Biden said.
"What did they do? They poured hundreds of millions of dollars and thousands of tons of weapons into anyone who would fight against Assad except that the people who were being supplied were al-Nusra and al-Qaida and the extremist elements of jihadis coming from other parts of the world.
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