Turkish Prime Minister Binali Yildirim urged Athens to return eight soldiers accused of taking part in an attempted coup last July.
Yildirim made the statement in Athens where he met Greek Prime Minister Alexis Tsipras.
"The Greek justice system is completely independent and we have to respect its decisions, regardless of whether we agree with them or not," Tsipras said, speaking at a joint press conference shortly after their meeting ended.
"Politically speaking, we are clearly against the coup attempt, and coup plotters are not welcome in our country".
In July 2016, eight former Turkish soldiers, fled their country hours after a defeated coup attempt, in which 250 people were martyred, and arrived in Greece on a highjacked Blackhawk helicopter. They then sought asylum in Greece.
Ankara has repeatedly requested that the men be extradited in Turkey but the Greek courts have turned down the request twice, after a series of hearings and appeals.
Yildirim responded to Tsipras by saying that even though he acknowledged that judicial and executive branches were different, "we believe the [alleged coup plotters] should be returned".
Cyprus solution needed
On the Cyprus issue, both leaders agreed it was imperative to find a solution that would "safeguard the future and the security of both Turkish and Greek Cypriots", Yildirim said.
"We want a fresh start with a new vision," Yildirim told reporters regarding EU-Turkish relations.
He added the migration deal signed in 2015 had helped refugee flow considerably decrease, despite the fact that Europe had not yet fulfilled its part of the deal.
"Visa liberalisation for the Turks and financial support towards the country," have not yet been properly addressed, the Turkish leader added.
Greece supports Turkey's EU accession course, which can help resolve issues and stimulate further cooperation, Tsipras said.
After his meeting with Tsipras, Yildirim met Greek President Prokopis Pavlopoulos. He is scheduled to meet opposition leader Kyriakos Mitsotakis and to depart for Komotini where he will take part in an Iftar, fast-breaking, dinner.