Greek President Karolos Papoulias is preparing to hold talks with party leaders in an attempt to create an emergency government.
The move comes after the country's socialists became the third party to fail to form a coalition.
If the president's bid fails, fresh elections will have to be held, probably next month.
Last Sunday, voters backed parties opposed to Greece's bailout deal that requires deep budget cuts.
Greece's political turmoil has raised the possibility that it could default on its debts and be forced out of the eurozone.
The president is expected to try to pressure parties into a government of national salvation - but the BBC's Mark Lowen in Athens says he is unlikely to succeed.
The process could take days.
Evangelos Venizelos, leader of the socialist Pasok party, abandoned efforts to form a new government on Friday and said he would meet the president on Saturday morning.
He had held talks with centre-right New Democracy leader Antonis Samaras, whose party came first in the election, but could not find a third partner to give them a majority in parliament.
"I hope that during the negotiations chaired by Mr Papoulias everyone will be more mature and responsible in their thinking," Mr Venizelos said.
New Democracy also failed to form a coalition earlier in the week, as did the left-wing bloc Syriza, which came second in the election.
Syriza firmly rejects the terms of the EU-IMF bailout, which requires tough austerity measures in return for loans worth a total of 240bn euros (£192bn; $310bn).
Its leader, Alexis Tsipras, said on Friday he could not join any coalition that intended to implement the bailout deal.
"The rejection of this plan does not come from Syriza but was given by the Greek people on the night of the election," he said.
"The bailout austerity has already been denounced by the Greek people with its vote, and no government has the right to enforce it."
Analysts say Syriza could be hoping for fresh elections after one opinion poll put them in first position in any new ballot, albeit without an overall majority.
Sunday's election saw a backlash against Pasok and New Democracy, which had formed the outgoing coalition and had agreed the terms of the bailout.
The once-dominant Pasok, which was seen as the architect of austerity, came third with just 41 seats in the 300-seat parliament.
The Greek crisis is continuing to create unease is global financial circles.
The Fitch ratings agency warned that if Greece did leave the euro, it would probably place all 16 remaining euro nations' sovereign ratings on "rating watch negative" - meaning they would be in danger of being downgraded.
"A Greek exit would break a fundamental tenet underpinning the euro - that membership of EMU (Economic and Monetary Union) is irrevocable," Fitch said.
EU monetary affairs chief Olli Rehn said Greece had to abide by its bailout terms.
"Greece systemically lived beyond its means for a decade... It is simply not sustainable and therefore Greece has had to take firm action to restore its economic competitiveness and sustainable public finances," he said.