Gokcek, mayor for over 22 years and one of the most senior figures in the ruling Justice and Development Party (AKP) of President Recep Tayyip Erdogan, also told AFP in an interview he feared Turkey was entering a period where top officials could be at risk of assassination.
Turkey has squarely blamed Gulen for masterminding the rebellion, saying he assiduously built up a "parallel state" with followers in all institutions. From his secluded compound in Pennsylvania, the preacher has denied the charges.
Turkish authorities have launched a sweeping nationwide purge of suspected Gulen supporters since the coup, dismissing more than 50,000 people from their jobs and detaining more than 18,000.
With some officials now even alleging that Washington could have had a hand in the putsch, Ankara wants the United States to send back Gulen to face trial in the country he left in 1999.
"For America to prove it is not behind the coup, there is only one thing to do, deliver (him) to Turkey," Gokcek told AFP in the capital.
He claimed that the US had already given "signals" it was involved in the coup after a top American general expressed concern that many of Washington's former Turkish military interlocutors were now in jail.
"How will it be known whether America is or is not involved in this business? If they deliver (Gulen) there is no problem. But if they don't the United States will not escape from the dock."
Gokcek, who had previously suggested Gulen was hypnotising people, expressed bewilderment that Washington had tolerated the cleric's presence and allowed his foundations to open up schools in the US.
"For America to tolerate this, it seems there are connections to FETO," he said, referring to what Turkey calls the Fethullah Terror Organisation (FETO).
The US State Department has rejected suggestions it had any hand in the coup as "ludicrous".
Gokcek said Gulen's role in Turkey went back to the premierships of Bulent Ecevit in the 1970s and admitted the AKP had mistakenly formed an alliance with his similarly Islamic-leaning supporters when it first came to power in 2002.
"But their biggest aim was to use us and to get their own people inside the army," he said.
Gokcek, one of the most outspoken senior figures in the AKP who regularly updates 3.4 million followers on Twitter with his views, said there would no longer be a risk of coups in Turkey but rather of assassinations.
"Politicians will be at risk of assassinations... Of course I have increased my security," he said.
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