The U.S. Embassy in Ankara issued a statement questioning the commitment of Turkey to the security of U.S. mission's facilities and personnel.
The dispute erupted last week when Turkey arrested a Turkish employee of the American consulate on suspicion of links to the group blamed for last year's failed coup.
The U.S. suspended visa services for visitors from Turkey in a sharp escalation of tensions between the countries, sending the lira down as much as 1 percent against the U.S. dollar.
"But our ministers, parliament speaker and myself did not accept and will not accept his request because we do not see him as a representative of the United States", Erdogan told a televised news conference during a visit to Belgrade.
Erdogan's spokesman Ibrahim Kalin previously pointed to a phone call made from the Istanbul consulate to a key suspect on the night of the coup.
The move followed the arrest on October 4 of a Turkish national who works at the USA consulate in Istanbul for alleged involvement in the July 2016 coup attempt on Turkish President Recep Tayyip Erdogan.
Mr Erdogan accuses Mr Gulen of instigating the unrest - a charge the cleric denies.
"The US should evaluate one thing: how did those agents leak into the consulate?"
The U.S. had said the arrest and allegations were "wholly without merit".
Turkey's state-run media quoted the Attorney General as saying an employee of the consulate in Istanbul, "who has no diplomatic immunity, has been invited to the prosecutor's office for testimony and his wife and children were taken into custody in Amasya, Turkey".
It is believed that the issue emerged right after a U.S. consulate worker in Istanbul was held last week on suspicion of links to a cleric blamed for last year's failed coup in Turkey.
In addition, although Erdogan was strongly opposed to the Kurdish independence referendum that took place on September 25 in Iraqi Kurdistan, and "threatened to do a lot of things" against Erbil, "he has done nothing so far at least", Jenkins highlighted.
The U.S. Embassy in Turkey on Sunday announced the suspension of non-immigration visas for Turkish nationals following the arrest of a Turkish employee at the U.S. consulate in Istanbul.
Pro-government media have rounded on the United States, with the Yeni Safak daily describing the USA as "not an ally but an enemy".
But that is not all: Ankara has yet to persuade Washington to extradite Gulen although 16 months have already passed since the attempted coup, he highlighted.
Ankara is also angry over the arrest in the United States of Mehmet Hakan Atilla, deputy chief executive at state lender Halkbank, and the Iranian-Turkish businessman Reza Zarrab. "Does this fit our alliance or friendship?" "Neither side is willing to step back", he said.