Kerry said the United States would entertain an extradition request for Gulen, but Turkey would have to present "legitimate evidence that withstands scrutiny".
"On the grounds of suspicion, he can be easily extradited".
On Tuesday, Turkey ordered 1,577 deans at state and private universities to resign, a day ahead of an "important" announcement from Erdogan. In the latest moves on Monday, Turkey's police force was purged.
Since the failed putsch on Friday, around 9,000 people including police and government officials have been sacked.
U.S.-based cleric Fethullah Gulen, whose followers Turkey blames for a failed coup, is shown in still image taken from video, speaks to journalists at his home in Saylorsburg, Pennsylvania July 16, 2016.
In his statement to prosecutors, Ozturk denied he was the coup ringleader.
He told USA broadcaster CNN that he narrowly escaped death after coup plotters stormed the resort town of Marmaris where he was on holiday.
The vague directive doesn't identify specific media outlets, leaving it open for interpretation.
"From now on, Gulen issue would be the most important topic of the Turkish-American relations", Tanis said, adding that the issue would dominate discussions between officials of the two countries.
Meanwhile, Mr Erdogan has urged citizens to remain on the streets even after the defeat of the coup attempt, in what the authorities describe as a "vigil" for democracy.
Binali Yildirim, Turkey's prime minister and longtime loyal aide to president Recep Tayyip Erdogan, has announced that over 7,500 people have been arrested in connection with the coup. They may also include new evidence that has emerged from the current investigation. "There was a list of people who were suspected of conspiring to stage a coup", the official said.
The agency said Erdogan's Air Force adviser, Lt. Col. Erkan Kivrak, had been detained at a hotel where he was vacationing in Turkey's southern province of Antalya.
Gulen has lived in self-imposed exile in the US since 1999 when he fled Turkey amidst accusations of extremist Islamist activities. According to the Anadolu news agency, Greek Prime Minister Alexis Tsipras called Yildirim on Sunday night and confirmed that the extradition process had begun.
Tolga Tanis, a local daily Hurriyet columnist, believes that a crisis between Ankara and Washington will erupt over Gulen case after the coup attempt. He said the July 15 victory over the plotters was "epic" and that no coup in the history of Turkey had been as brutal as the one that the elected government survived. The bank's Monetary Policy Committee said it has reduced its overnight marginal funding rate from 9 percent to 8.75 percent. All other interest rates were left unchanged. "Because those who attempt a coup in this country must pay". Despite a modest rally Tuesday, Turkish stocks are still way down from pre-coup levels, as is the Turkish lira.
The coup saw 294 people killed in the chaos, and 3,000 judges have been suspended.
"Reintroduction of the death penalty would be in breach of Turkey's obligations under worldwide human rights law - a big step in the wrong direction", he said.