"There has always been an open channel of communication between the countries, and therefore there is no need for a new beginning," said Davutoglu, "We are willing to resume our role as mediators between Israel and Syria."
As for the Israeli-Palestinian conflict, the Turkish FM called on the sides to "launch a new era."
Earlier Monday, Ben-Eliezer said Israel wants to improve and strengthen ties with key Muslim ally Turkey, which have soured over Israel's military operation in the Gaza Strip.
Ben-Eliezer, the first Israeli minister to visit Turkey since relations hit a low last winter, also said Turkey can help "solve problems in the Middle East."
Muslim but secular Turkey has long been Israel's closest ally in the Muslim world and mediated months of indirect talks between Israel and Syria last year.
But ties deteriorated steeply after Prime Minister Recep Tayyip Erdogan, who heads an Islamic-oriented government, strongly criticized Israel's military operation in Gaza and complained of the heavy Palestinian death toll.
The Turkish-mediated indirect talks between Syria and Israel were put on hold over the Gaza offensive.
Ben-Eliezer told a group of Turkish business representatives that Israel "wants to continue and strengthen relations, we want to clean the cloudy situation."