He said he had sent the US evidence of Fethullah Gulen’s criminal activities – allegations the cleric denies – in support of an extradition bid.
Mr. Yildirim insisted that his country was governed by the rule of law.
Thousands of soldiers, police and officials have been detained or sacked since Friday’s coup attempt.
President Recep Tayyip Erdogan has again refused to rule out reinstating the death penalty for coup plotters if it is approved by parliament.
The EU has warned that such a move would see the end of accession talks to the bloc.
For now, at least, that seems not to worry President Erdogan, who is seizing the opportunity to tighten his grip, reports the BBC’s Turkey correspondent, Mark Lowen.
Prime Minister Yildirim was speaking after meeting the leader of the main opposition CHP party.
He warned people not to act out of a spirit of revenge in the wake of Friday’s failed military takeover, saying that would be “unacceptable” but whoever had acted against the law would be punished.
“Today we need unity,” he said.
The interior ministry has dismissed almost 9,000 police officers as part of a purge of officials suspected of involvement in the coup attempt.
That followed the arrest of 6,000 military personnel and suspension of almost 3,000 judges over the weekend.
Many of those accused of involvement are closely linked to the ruling apparatus. -BBC