Greece's socialist leader, Evangelos Venizelos, says efforts to form a coalition government have failed.
Venizelos, who was the last of three party leaders to try to reach an agreement, said he would hand the mandate back to President Carolos Papoulias on May 12.
He made the comments after meeting with Alexis Tsipras, whose antiausterity Syriza party came second in last weekend's election.
Tsipras said he would not join any government that intended to continue implementing the terms of Greece's international bailout agreement.
The May 6 inconclusive elections have boosted the standing of antiausterity parties, which want to abolish Greece's bailout deal with the EU and the International Monetary Fund (IMF).
Athens must pass new austerity measures worth 14.5 billion euros by June in order to qualify for the next installment of the bailout.
New elections would have to be called in June.
EU president Herman Van Rompuy meanwhile called on Greece's political leaders to make responsible decisions to resolve the country's debt crisis. Van Rompuy, on a visit to Denmark, said he was "concerned about the situation in Greece" and appealed "to the sense of national responsibility, of all political leaders."
Based on reporting by AFP and AP