Kerry said he told Turkish Foreign Minister Mevlut Cavusoglu that the extradition should be demanded on “legal grounds”.
He also voiced support for Turkey as a NATO ally and said the country is “a key partner to EU”.
“We stand squarely on the side of elected leadership,” Kerry said.
However, he called on Turkey to respond to the coup supporters, who sought to overthrow the democratically elected government, within “the standards of respect for [the] nation’s democratic institutions and rule of law”.
“We will certainly support bringing the perpetrators of the coup [to] justice but we also caution against a reach that goes beyond that,” Kerry said.
Mogherini also stressed that legitimate institutions needed to be protected and that the EU was the first “to stress the need for having the legitimate institutions protected against the attempt of coup”.
In Washington, the White House denied allegations that it is harboring Gulen, saying that such claims are “factually incorrect”.
Spokesman Josh Earnest said when Turkey formally puts forward its extradition request, it “will be carefully considered by the U.S. government consistent with the Extradition Treaty that’s been on the books more than 30 years now.”
Friday’s failed coup was organized by followers of Gulen, who is accused of pursuing a long-running campaign to overthrow the government through supporters within the Turkish state, particularly the military, police, and judiciary.
Described as the “parallel state”, the Gulenist Terror Organization/Parallel State Structure (FETO/PDY) is also behind the December 2013 corruption investigation into senior Turkish government officials.
The FETO/PDY is headed by Gulen, who runs a network of schools and commercial enterprises in Turkey and around the world, including Harmony in Texas.
Gulen is also the main suspect in the December 2013 investigation of mass wiretapping, which targeted more than 7,000 people.
Last October, an Istanbul court issued an arrest warrant for Gulen after approving a 1,453-page indictment, charging him with “attempting to overthrow the government of the Republic of Turkey or obstructing it from conducting its duties by force”.
Gulen is also accused of “establishing and masterminding an armed terrorist organization” and “obtaining state information — which needs to be kept secret for the security of the state and its domestic and foreign benefits — to be used for political espionage.”
Since early 2014, investigations into the parallel state have seen hundreds of civil servants, including police and public prosecutors, arrested or reassigned.