Justice minister Bekir Bozdag said some 6,000 people have been held. The figure includes about 2,700 judges and prosecutors.
The government has also suggested it could bring back the death penalty for those involved in the coup.
Mr Bozdag also said he was confident that the United States would return Islamic cleric Fethullah Gulen to Turkey.
Turkish president Recep Tayyip Erdogan has blamed Mr Gulen and his followers for Friday night’s failed coup.
However, Mr Gulen has denied any involvement in or knowledge about the attempted coup. The US said it will look at any evidence Turkey has to offer against Mr Gulen, and decide accordingly.
Mr Bozdag said: “The US would weaken itself by protecting him. It would harm its reputation. I don’t think that at this hour, the United States would protect someone who carried out this act against Turkey.”
Speaking at a funeral in Istanbul yesterday, Mr Erdogan pledged to “clean all state institutions of the virus” of Gulen supporters.
Mr Erdogan has also spoken with Russian president Vladimir Putin. A statement from Mr Erdogan’s office yesterday said that Mr Putin said Moscow stood by “Turkey’s elected government” and expressed his good wishes to the Turkish people.
It said the two leaders – who recently restored diplomatic relations following Turkey’s downing of a Russian warplane – also agreed to meet face to face next month.
At least 290 people were killed and more than 1,400 were wounded in the coup. Government officials said at least 104 conspirators had been killed.
Prayers were being read simultaneously from Turkey’s 85,000 mosques to rally the country to defend its democracy and honour those who died.
Three of the country’s top generals have already been detained, alongside hundreds of soldiers.
The government also dismissed nearly 3,000 judges and prosecutors from their posts, while investigators were preparing court cases to send the conspirators to trial on charges of attempting to overthrow the government.
The botched coup, which saw warplanes fly over key government installations and tanks roll along streets in major cities briefly, ended hours later when loyal government forces regained control of the military and civilians took to the streets in support of Mr Erdogan.
Chanting, dancing and waving flags, tens of thousands of Turks – some walking hand in hand with their children – marched through the streets late on Saturday and early on Sunday to defend democracy and support the president.
Crowds chanted “Fethullah will come and pay”, “Allah is Great” and “We want the death penalty”. Mr Erdogan said that in democracies “you cannot push the wish of the people to one side” but also said “we are not after revenge”.
Rather than toppling Turkey’s strongman president, the attempted coup appears to have bolstered Mr Erdogan’s popularity and grip on power.
Gozde Kurt, 16, a student who had joined a rally in Istanbul, said: “Just a small group from Turkish armed forces stood up against our government … but we, the Turkish nation, stand together and repulse it back.”
The Yeni Safak newspaper used the headline “Traitors of the country”, while the Hurriyet newspaper declared “Democracy’s victory”.
Turkey’s prime minister, Binali Yildirim, said the perpetrators of Friday’s failed coup “will receive every punishment they deserve”.
However, the government crackdowns raised concerns over the future of democracy in Turkey, which has long prided itself on its democratic and secular traditions.
Secularism was first introduced in 1928, and bolstered by the reforms of Turkey’s first president, Mustafa Kemal Atatürk. The current Constitution of 1982 does not recognise an official religion.
Mr Erdogan’s survival despite the coup has turned him into a “sort of a mythical figure” and could further erode democracy in Turkey, said Soner Cagaptay, director of the Turkish research programme at US think-tank the Washington Institute.
He said: “It will allow him to crack down on liberty and freedom of association, assembly, expression and media in ways that we haven’t seen before.”
The coup attempt started late on Friday night with tanks rolling onto the streets of Ankara and Istanbul as the president was on a seaside break.
Explosions and gunfire erupted throughout the night. It quickly became clear, though, that the military was not united in the effort to overthrow the government.
In a TV interview conducted via a smartphone video app, Mr Erdogan urged his supporters into the streets to confront the troops and tanks, and forces loyal to the government began reasserting control.
Before the chaos, Turkey – a Nato member and key ally for the western in the fight against Islamic State – was wracked by political turmoil that critics blamed on Mr Erdogan’s increasingly authoritarian rule. He has cracked down on dissidents, restricted the news media and renewed fighting with Kurdish rebels.