More than 100 judges and military officers were detained Sunday over alleged ties to the attempted military coup, the Turkish-run Anadolu news agency and CNN Turk reported.
Erdogan has accused Gulen, a reclusive Muslim cleric living in self-imposed exile in a mountain town in Pennsylvania, of being behind Friday's bloody putsch attempt.
USA officials were working with Turkish officials to resume air operations quickly, the Pentagon said.
Obama conferred with his national security and foreign policy advisers on Saturday morning and reiterated his support for the "democratically-elected, civilian" government of Turkey.
The US has warned Turkey against public insinuations of American involvement in a failed military coup, saying such claims were utterly false and harmful to their relations.
Thousands took to the streets to demonstrate in defiance of the attempt to overthrow the government.
Erdogan also telephoned main opposition Republican People's Party (CHP) leader Kemal Kilicdaroglu and Nationalist Movement Party (MHP) Chairman Devlet Bahceli, thanking them for their support for the ruling Justice and Development Party (AK Party), democracy, and the national will, and for not lending credence to the coup attempt, the source said, speaking on condition of anonymity, due to restrictions on speaking to the media. Prime Minister said Turkey has already delivered its request of extradition for Gülen.
Turkey accuses Gulen of leading a group called the "Fethullahci Terror Organisation (FETO)" that has created a parallel state. Erdogan called on the Turkish people to flood the streets in a show of support for his embattled government.
In the United States, a lawyer hired by the Turkish government has lodged numerous accusations against a network of about 150 publicly funded charter schools started by followers of Gulen, whose philosophy blends a mystical form of Islam with staunch advocacy of democracy, education, science and interfaith dialogue.
Prime Minister Binali Yildirim said the perpetrators of Friday's failed coup "will receive every punishment they deserve", and the government said it would take steps toward extraditing a U.S.-based cleric it accused of fomenting the uprising.
US Secretary of State John Kerry said of any extradition, that Turkey should "present us with any legitimate evidence that withstands scrutiny". Nearly a similar number of judges and prosecutors have been dismissed.
Before the weekend's chaos, Turkey - a North Atlantic Treaty Organisation member and key Western ally in the fight against the Islamic State group - had been wracked by political turmoil that critics blamed on Erdogan's increasingly authoritarian rule. Gulen denies the charges. "The United States is entering into a corollary situation to what we have with Pakistan".
The Incirlik air base near Adana in the southeast of the country was running on internal power sources after a loss of commercial power to the base, the Pentagon said.
"There is a slight chance, there is a possibility that it could be a staged coup", Gulen told reporters through a translator in Pennsylvania, where he resides.