The Obama administration appeared to be on a collision course with a key ally after Turkish President Recep Tayyip Erdogan demanded the extradition of an aging cleric living in the U.S. whom he accused of orchestrating Friday’s failed coup.

Turkish government officials on Saturday said Turkey would view the U.S. as an enemy if the Obama administration doesn’t hand over Fethullah Gulen, a reclusive but influential Turkish cleric who has lived in self-imposed exile in Pennsylvania for nearly two decades.

“Any country that stands behind him is no friend of Turkey, is engaged in a serious war with Turkey,” Prime Minister Binali Yildirim said from his office in Ankara. Mr. Erdogan, in televised remarks from Istanbul, made a personal appeal to President Barack Obama.

“Today, after this coup attempt, I’m once again calling on you, I’m saying: Extradite this man in Pennsylvania to Turkey now,” he said.

Secretary of State John Kerry responded sharply, telling Turkish Foreign Minister Mevlut Cavusoglu in a phone call Saturday that “public insinuations or claims about any role by the United States in the failed coup attempt are utterly false and harmful to our bilateral relations,” said John Kirby, the State Department’s spokesman.

Mr. Kirby said Mr. Kerry told his Turkish counterpart that the U.S. would be willing to assist authorities in Ankara investigating the uprising but said the U.S. also “urged restraint by the Turkish government and respect for due process—and its international obligations—as it investigates and uncovers additional information about those involved.”

The unusual exchange exposed longstanding tensions between the two governments over a range of issues spanning civil liberties, governance and the international campaign against the Islamic State extremist group, which were forced to the surface in the wake of the coup attempt.

Turkish officials, describing the same conversation, said Mr. Kerry spoke with Mr. Cavusoglu to discuss the legal process regarding Mr. Gulen’s extradition to Turkey. An extradition request is expected to be submitted to the Justice Department, although Mr. Kerry said Saturday the U.S. hadn’t received one yet. He didn’t rule out a possible extradition, telling reporters in Luxembourg on Saturday that the U.S. would weigh any evidence provided by Ankara.

The Justice Department refused to comment.

A onetime ally of Mr. Erdogan in fighting to separate the military from politics, Mr. Gulen maintains a significant following in Turkey, preaching a message of tolerance. His residence in the U.S. has long been a point of tension for Turkey.

Mr. Gulen, in a rare press briefing at his compound in Saylorsburg, Pa., denied he played any role in the coup and charged Mr. Erdogan with staging it himself to strengthen his power and weaken the military’s. He said he believes he would face persecution if forced to return to Turkey.

In a separate statement emailed by a spokeswoman, Mr. Gulen denounced the overnight coup attempt and said he and his followers have long rejected interference by the military in civil affairs.