Speaking in Washington, President Barack Obama said there was a legal process for extradition and encouraged Turkey to present evidence.
In a sign of increasing tension, Turkey said it was dispatching its justice and interior ministers to the United States next week to push for the extradition of the cleric, Fethullah Gulen.
The two NATO countries are allies in the fight against the Islamic State group; American military jets have been flying missions against extremists in Iraq and Syria out of the Turkish air base at Incirlik.
Meanwhile, Turkey's prime minister, Binali Yildirim, warned that coup plotters still at large might stage attacks, saying there is "a remote chance some madmen might take action, acting out of a sense of revenge and defeat."
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Turkey has launched a sweeping crackdown following the failed July 15 insurrection, declaring a three-month state of emergency and detaining or dismissing tens of thousands of people in the military and other state institutions.
Turkey alleges that the coup attempt by some military units was conceived by Gulen, who has lived in self-imposed exile in Pennsylvania since the late 1990s. Gulen has denied any prior knowledge of the coup attempt.
Yildirim criticized the United States for failing to hand over the cleric, a former ally of President Recep Tayyip Erdogan, Turkey's most dominant political figure for more than a decade.
"Stop standing up for savages who run over citizens with tanks, who strafe people from land and the air," Yildirim said.
In his comments today, Obama said any reports that the United States had previous knowledge of the coup attempt or has been anything other than supportive of Turkey's government are "unequivocally false.