However, Serdar Kılıç also expressed frustration with the slow pace of those proceedings, and suggested the US administration could take further steps outside the extradition process to censure Gulen.
"They are working more seriously on the issue, and there have been a number of contacts on that regard," Kılıç noted, specifically referencing a visit by the Turkey's foreign minister and justice minister, who presented "additional files" of evidence against Gulen to US Attorney General Jeff Sessions in May.
"But if you ask my personal opinion," he added, "it's not moving as fast as the Turkish public opinion would like it to move."
Kılıç suggested the Trump administration should put "limitations" on Gulen's activities in the United States, including his ability to give interviews and to travel.
The Department of Justice declined to comment on the status of Turkey's extradition request with an official telling CNN: "As a general policy the department doesn't comment on extradition matters."
Gulen, 76, has been living in self-imposed exile in Pennsylvania since 1999 and runs a lucrative network of charter schools.
The imam also leads the Hizmet movement, with supporters in Turkey. He once backed Turkish President Recep Tayyip Erdogan, but the pair have since become bitter rivals.
Gulen has denied any connection to the coup attempt.
The Turkish government accuses him of being behind the coup attempt last summer, which resulted in the deaths of nearly 300 people.
In the aftermath, the government imposed a nationwide state of emergency and arrested tens of thousands of alleged Gulen supporters, known as Gulenists.
That state of emergency remains in effect one year later.
The decision to lift the state of emergency, said Kılıç, "will be taken after the proper investigations are concluded and we reach to the point that the Fethullah Gulen terrorist organization is cleansed from the Turkish state structure, and that they're not going to pose a similar threat in the future to the Turkish democracy, and to the Turkish people and to the future of the Turkish state."