Erdogan made the remarks during his meeting with Tsipras on the sidelines of the Belt and Road Forum in Beijing.
The Turkish president told the Greek premier the non-extradition was “a problem” between the two countries and termed the ongoing stay of the ex-soldiers in Greece as “incorrect”.
He also asked the prime minister to take necessary steps to resolve the issue, the source said.
On May 4, a Greek appeals court rejected the extradition of two former Turkish soldiers, joining six others whose extradition was similarly denied.
The ruling meant that Greek courts had rejected the extradition of all eight ex-soldiers who fled Turkey soon after the collapse of last July’s coup attempt, which left 249 people martyred and nearly 2,200 injured.
The former soldiers are accused of being involved in the foiled coup, which Turkey says was masterminded by the Fetullah Terrorist Organization (FETO) and the group’s leader, U.S.-based Fetullah Gulen.
The Turkish government has repeatedly requested the extradition of the eight alleged coup plotters, promising they would get fair trials in Turkey.
Also, Erdogan and Tsipras agreed during their meeting that talks to resolve the Cyprus issue should continue with a "positive" agenda, the source added.
Erdogan is in China to attend the Belt and Road Forum that will officially begin in the capital Beijing on Sunday.
The Belt and Road Initiative was proposed by Chinese President Xi Jinping in 2013.
According to China’s official Xinhua news wire, the goal of the initiative is to build a trade and infrastructure network along ancient trade routes such as the Silk Road that would connect Asia with Europe, Africa and beyond.
Erdogan also met Pakistani Prime Minister Nawaz Sharif on the sidelines of the forum on Saturday. Both leaders agreed to further accelerate their efforts for the signing of a comprehensive free-trade agreement between Turkey and Pakistan.
The Turkish president also explored options to bilateral trade volume with Hungarian Prime Minister Victor Orban.
Later, Erdogan met Jinping following which three treaties between the two countries were signed, including one on the extradition of criminals.
He also held an hour-long meeting with UN Secretary General Antonio Guterres.