Turkey is pressing the US to extradite Mr Gulen and the issue was raised during a phone call between US President Barack Obama and President Erdogan on Tuesday, the White House said.
Spokesman Josh Earnest said a decision on whether or not to extradite would be made under a treaty between the two countries.
In a separate development Turkey has barred access to the WikiLeaks website soon after it posted about 300,000 emails sent by President Erdogan's AK Party dating from 2000 to early July 2016.
Wikileaks said that although the documents were obtained before the attempted coup, the date of their publication was brought forward "in response to the government's post-coup purges".
The source of the emails was not linked to the coup plotters or to a rival political party or country, WikiLeaks said.