Public prosecutor of Ankara, which carries out the investigation into the failed coup attempt, demanded the arrest of Gulen, who has been in self-imposed exile in Pennsylvania since 1999.
It is also said that the Sakarya prosecutor general's office has made a decision to detain Sukur's father Sermet Sukur, although this is also unconfirmed.
Sukur is Turkey's leading goalscorer with 51 goals from 112 appearances and is best known internationall for his performances at the 2002 World Cup in Japan/Korea where Turkey finished third.
Yıldırım also discussed the ongoing fight against the FETO.
-Turkish relations is dependent on the United States cooperating with their request to extradite Gulen.
Gulen is accused of ordering the July 15 coup during which a group within the military tried to remove President Recep Tayyip Erdogan from power.
The request has been submitted to Turkey's Justice Ministry on Saturday for it to be relayed to USA authorities.
On July 15 evening, Turkish authorities said a military coup attempt took place in the country.
Yildirim said he believed there would be a "positive outcome" with Washington on the extradition, Anadolu said.
More than 200 people were killed in the attempted overthrow of Erdoğan and his government.
He said that 76,597 people were suspended from work and 4,897 dismissed in a speech at the Cankaya Palace in Ankara, Anadolu news agency reported.
The Turkish PM said the United States attitude over the coup attempt had a positive shift.
But he resigned in 2013 after a vast corruption probe that targeted Erdogan and his inner circle, siding with the movement of Erdogan's arch-foe Gulen.
Speaking to Reuters on Saturday in Turkey's mainly Kurdish southeastern province of Diyarbakir, Selahattin Demirtas, head of Turkey's pro-Kurdish opposition, said Erdogan had allowed Gulen's followers to tighten their hold within state bodies.
Sukur is facing charges of "being a member of an armed terrorist organisation", according to officials.