Referring to the United States, Mr. Yildirim said, “we would be disappointed if our friends told us to present proof even though members of the assassin organization are trying to destroy an elected government under the directions of that person.” He added, “At this stage there could even be a questioning of our friendship.”

Mr. Kerry has said Turkey, as part of the extradition process, must provide evidence that withstands scrutiny in an American court — something analysts say Turkey does not have.

On Tuesday, Mr. Gulen again denied any involvement. “Turkish President Recep Tayyip Erdogan today once again demonstrated he will go to any length necessary to solidify his power and persecute his critics,” Mr. Gulen said in a statement. “It is ridiculous, irresponsible and false to suggest I had anything to do with the horrific failed coup. I urge the U.S. government to reject any effort to abuse the extradition process to carry out political vendettas.”

Turkish officials may be certain about Mr. Gulen’s actions and motives, but the nature of his movement has long confounded analysts and diplomats in Turkey, partly because the organization is opaque and individuals do not openly declare allegiance to it.

Mr. Jeffrey said it would have been hard for Gulen followers, as Islamists, to infiltrate the armed forces, which have been a stronghold of secularism in Turkey.

In diplomatic cable written in 2009, and made public by WikiLeaks, Mr. Jeffrey detailed how Mr. Gulen came to exile in the United States. He left Turkey in 1999 after being charged with plotting to overthrow the state. The charges, Mr. Jeffrey wrote, were based on a sermon Mr. Gulen had given in which he said, “our friends, who have positions in legislative and administrative bodies, should learn its details and be vigilant all the time so they can transform it and be more fruitful on behalf of Islam in order to create a nationwide restoration.”

Mr. Gulen was later acquitted, in absentia, on all charges.

Jenny White, a professor at the Stockholm University Institute for Turkish Studies who has studied the Gulen movement, said it is centered on a worldwide network of secular schools. The goal, she said, is to create a “golden generation of young people who are educated in science, but have Muslim ethics.”