The Turkish embassy in Washington said the allegations were “utterly false, ludicrous and groundless”.
Ex-CIA director James Woolsey revealed an alleged $15m (£11.5m) plan to remove cleric Fethullah Gulen in March.
Turkey accuses Mr Gulen of being behind the failed July 2016 coup.
In a statement, the embassy said: “The Turkish people expect the immediate extradition of Fethullah Gulen from the United States to Turkey, so that he can stand trial.”
They added: “All allegations that Turkey would resort to means external to the rule of law for his extradition are utterly false, ludicrous and groundless.”
Fethullah Gulen: Powerful but reclusive Turkish cleric
Michael Flynn: Former US national security adviser
Turkish President Recep Tayyip Erdogan has repeatedly called for Mr Gulen’s extradition from the US, where he lives in Pennsylvania.
President Trump’s former national security adviser, Michael Flynn, is accused of meeting Turkish officials in September 2016 to discuss the removal of Mr Gulen.
My Flynn’s lawyer, Robert Kelner, said the claims were “outrageous”.
According to The Wall Street Journal, Mr Flynn – who quit his post after misleading the White House about meeting a Russian envoy before Donald Trump took office – discussed having Mr Gulen transported on a private jet to the Turkish prison island of Imrali.
It also reported that a payment was offered.