Davutoglu's Justice and Development Party (AKP) failed to secure enough votes in the June 7 election to form a government alone, for the first time since it came to power in 2002.
The AKP has 258 seats in the new parliament, the Republican People's Party (CHP) 132, and the Nationalist Movement Party (MHP) and pro-Kurdish People's Democratic Party (HDP) hold 80 apiece.
Davutoglu met today with CHP leader Kemal Kilicdaroglu but no agreement is expected this week during the Eid feast marking the end of holy Muslim fasting month of Ramadan.
He hailed "very sincere and friendly" talks with the CHP team.