Erdogan and the government have said that Gulen's followers in the military were responsible for the attempted take-over on Friday night and early Saturday morning.
Reuters quotes a security source as saying some 8000 police officers have been removed over possible links to the coup.
It is not clear how the post-coup purge will affect the judiciary, how the government will move to replace the dismissed judges and prosecutors, or where the trials for those detained would be held. Some of the arrested soldiers were shown in photographs with bruised faces, stripped to their underpants and handcuffed on the floors of police buses and a sports hall.
Canadian experts say the Turkish government is engaged in a witch hunt because it has arrested or fired thousands of judges and prosecutors following the weekend's failed coup.
Thousands of alleged coup plotters have been rounded up since Friday's failed putsch, in which more than 260 people were killed.
"A country that has the death penalty can't be a member of the European Union and the introduction of the death penalty in Turkey would therefore mean the end of accession negotiations".
The top USA diplomat said in Brussels that he told Turkish Foreign Minister Mevlut Cavusoglu "to make certain that in whatever portfolio and request they send us, they send us evidence, not allegations".
US Press Secretary Josh Earnest said the United States had not received any extradition requests for the cleric, but that it would evaluate one if it arrived under a 30-year-old extradition treaty.
"If you are in Turkey, restrict your movements, ensure that your travel documents are up to date, and keep abreast of the latest developments." it said.
The prime minister also said 50 coup backers and 1,491 government supporters were wounded in the unrest. "There is no excuse for any steps that take the country away from that". Turkey outlawed capital punishment in line with its efforts to join the European Union.
But on Tuesday Yildirim said action would be taken against Gulen's supporters.
They are urging Ankara not to take excessive measures when bringing the coup plotters to justice.
Turkish President Recep Tayyip Erdogan has reiterated that the death penalty is not off the table for those suspected of plotting a coup against his government, in an interview with CNN broadcast on Monday (July 28).
Gulen denies the charges - even raising the possibility the coup attempt may have been staged, but Erdogan has demanded his extradition from the U.S. It appears, he added, as if Turkey had "prepared" arrest lists of political opponents and was waiting for the right time to act.