He is also scheduled to meet Ahmet Necdet Sezer, the Turkish president, on Thursday before returning home.
Turkey's government has good relations with both Syria and Hamas, as well as Iran, with which it shares a long border.
Israel considers Iran a strategic threat, suspecting that it is building nuclear weapons, despite Tehran's denials. Iran's president has called for Israel to be wiped off the map, and Olmert hopes to enlist Turkey in accelerated efforts to keep Iran from going nuclear.
"Turkey is a bridge to other countries," Olmert said, hinting that Israel appreciates Turkey as a channel for communication to countries Israel does not have diplomatic relations with.
Turkey is likely to press Olmert to work with a new Palestinian government after last week's agreement by Hamas to join a national unity government with the Fatah faction of Mahmoud Abbas, the president.
Israel and the West have reserved judgment, insisting that any Palestinian government must recognise Israel, renounce violence and accept previous peace deals.
Israel, the US and Europe ban contact with Hamas, which they label a "terror" group.
Turkey is also expected to press Israel to accept peace overtures from Syria, but Israeli officials say Syria is interested in peace talks, not peace - as a way of ending its diplomatic isolation.
Before leaving Israel, Olmert acknowledged Turkey's potential, saying it is "an important country economically, diplomatically".