Erdogan has denounced the coup bid, which left more than 300 dead on all sides, as a treacherous bid to oust him from power devised from the US compound of his arch-enemy, exiled Islamic preacher Fethullah Gulen.
But with the authorities detaining over 7,500 people so far in a massive legal crackdown and sacking almost 9,000 people, Turkey's EU and NATO allies have urged Ankara to keep the rule of law in place.
General denies planning coup as Turkey presses purge
Erdogan's suggestion that the death penalty could be reinstated has sent shudders through Europe and sparked warnings such a move would be the nail in the coffin of its already embattled bid to join the EU.
An Ankara court late Monday placed under arrest 26 former generals suspected of planning Friday's attempted power grab, including former air force chief General Akin Ozturk, whom some Turkish media have painted as the mastermind of the plot.
The generals have now been put behind bars ahead of their trials, a date for which has not been sent.
They have been charged with crimes including seeking to overturn the constitutional order, leading an armed group and seeking to assassinate the president.
In his statement to prosecutors, Ozturk denied he was the coup ringleader.
"I am not the person who planned or led the coup. Who planned it and directed it, I do not know," state-run news agency Anadolu quoted him as saying.
Turkey's treatment of the suspects has alarmed its allies especially after some were paraded before the media and shown being subjected to rough treatment.
Anadolu published images of Ozturk and other suspects on the stairs inside the Ankara courthouse, staring blankly into the camera with their hands tied behind their backs.
Ozturk has looked tired and haggard in images published by state media, with one of his ears heavily bandaged.
Images have also emerged of government supporters physically attacking coup backers when they were detained.
"Nobody can have a feeling of revenge. This is unacceptable in a state governed by rule of law," Yildirim said after meeting secular opposition Republican People's Party (CHP) leader Kemal Kilicdaroglu.
The interior ministry said almost 9,000 people, including nearly 8,000 police but also municipal governors and other officials, had also been dismissed in a widening purge.
Turkish pilots who downed Russia jet part of coup plot: official
Turkey has blamed Friday's events on supporters within the military of Gulen, who Ankara accused of running a group it dubs the "Fetullahci Terror Organisation" (FETO).
Ankara has piled the pressure on Washington to extradite Gulen to face trial at home but US Secretary of State John Kerry said Monday he wanted to see evidence and not allegations.
Yildirim said Tuesday that Turkey had sent four dossiers to the United States over Gulen's alleged links to the overthrow plot.
The reclusive cleric said in an interview with several media outlets including AFP at his compound in the US state of Pennsylvania that he has "no concerns personally" about the extradition request.
"The rule of law reigns supreme here. I don't believe this government will pay attention to anything that is not legally sound," he said.
He called the putsch attempt "treason, a betrayal of the Turkish nation."
Turkey abolished the death penalty for all forms of crime in 2004 but the government now claims there is growing public pressure to reinstate it for the coup plotters.
Erdogan told CNN in his first media interview since the coup bid that he would approve any decision by parliament to reintroduce capital punishment.
"There is a clear crime of treason," he said.
But the EU -- which Turkey has for years tried to join in a stalled accession process -- warned of the consequences of such a move.
"No country can become an EU state if it introduces the death penalty," EU foreign affairs chief Federica Mogherini said.
Amnesty International said it was alarmed by the widening crackdown.
"Cracking down on dissent and threatening to bring back the death penalty are not justice," said John Dalhuisen, Amnesty's director for Europe and Central Asia.
Ex-Turkey air chief faces coup charges
Erdogan has remained in Istanbul ever since he dramatically flew back on Saturday to the city from the holiday resort of Marmaris where he was staying when the coup struck.
It was unclear when he would be returning to the capital Ankara. Every night since the coup he has spoken to supporters in the Istanbul district of Kisikli, where he has a home, urging them to maintain a "vigil" for democracy.
Erdogan told CNN his life had been in grave danger. "If I stayed (in Marmaris) 10, 15 minutes more, I would either have been killed or kidnapped and taken away by them."
Yildirim said 208 people were killed during the coup bid, including 145 civilians, 60 police and three loyalist soldiers. The military said 104 coup plotters were killed.
In an intriguing twist, a Turkish official said two Turkish pilots who played a role in the downing of a Russian plane in November that led to a crisis in ties between Moscow and Ankara are in custody over the failed coup.
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