Turkish officials on Tuesday, including Prime Minister Binali Yildirim, raised the pressure on the United States to hand over Mr. Gulen, promising to send dossiers of evidence of his role in the plot.
The White House said on Tuesday that it received an electronic file from Turkey on the matter, though it was unclear that it was a formal extradition request.
“The Department of Justice and the Department of State will review those materials consistent with the requirements of the extradition treaty between the United States and Turkey that’s been on the books for more than 30 years now,” Josh Earnest, the White House spokesman, said.
On Tuesday, Mr. Erdogan and Mr. Obama spoke by telephone, with Mr. Obama offering help to investigate the coup, but giving no indication in a statement by the White House of a willingness to promptly send Mr. Gulen back.
Mr. Yildirim said Turkey was intent on destroying the Gulen movement “by its roots.” And the government has moved quickly, raising concerns it is more interested in silencing all opposition than rooting out those behind the coup.
Nearly 35,000 members of the military, police and judiciary have either been arrested or dismissed.
On Tuesday, the government suspended more than 15,000 members of the Education Ministry, forced more than 1,500 university deans to resign and revoked the licenses of 21,000 private schoolteachers.
All of them, officials said, are suspected of having some link to Mr. Gulen.
The Turkish military, in a statement, blamed the “Fethullah Terrorist Organization” for the coup plot, and said the plotters had held at gunpoint the military’s chief of staff, demanding that he sign a document supporting the coup, which he refused to do.