The leader of Greece's socialist party, Evangelos Venizelos, has abandoned efforts to form a new government.
Mr Venizelos, the third leader to try to forge a coalition since Sunday's inconclusive elections, said he would now meet the president in a last-ditch effort to avoid fresh polls.
There has been no breakthrough in Mr Venizelos's talks with other parties.
Greece is deeply divided over budget cuts demanded in return for a bailout by the EU and the IMF.
The country's debt crisis has raised the possibility it could default and be forced out of the eurozone.
Following talks with other party leaders on Friday, Mr Venizelos told reporters: "I am going to inform the president of the republic tomorrow and I hope that, during the meeting with Carolos Papoulias, each party will assume its responsibilities."
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 country would then be facing the prospect of fresh elections, and this would be a leap into the unknown, our correspondent adds.
Sunday's election saw a surge in support for parties opposed to the terms of the bailout, as well as a backlash against Pasok and the conservative New Democracy party - which had formed the outgoing coalition.
Pasok, which was seen as the architect of austerity, came third with just 41 seats in the 300-seat parliament.
On Friday Mr Venizelos held talks with New Democracy leader Antonis Samaras, whose party came first in the election.
After the meeting, Mr Samaras told his MPs: "We are fighting for a government to exist - and there is still hope this can happen."
However Pasok and New Democracy between them have 149 seats, two short of a majority, and Mr Venizelos was unable to find a third coalition partner.
The Democratic Left party refused to join the coalition, as did the far-left Syriza party, which rejects the terms of the bailout.
Earlier this week, both Mr Samaras and Syriza leader Alexis Tsipras also tried and failed to put together a governing coalition.
The political deadlock has brought warnings from European leaders that debt-laden Greece could be thrown out of the euro if it does not stick to tough spending cuts and economic reforms.
On Friday, German Foreign Minister Guido Westerwelle told parliament: "The future of Greece in the eurozone lies in the hands of Greece.
"We want to and we will help Greece, but Greece has to be ready to accept help. If Greece strays from the agreed reform path, then the payment of further aid tranches won't be possible. Solidarity is not a one-way street," he said.
German Finance Minister Wolfgang Schaeuble told the Rheinische Post newspaper that Europe "won't sink" if Greece leaves the eurozone.
Athens is due to approve fresh budget cuts worth 14.5bn euros (£11.6bn; $18.8bn) next month, in return for financial help from the EU and IMF worth a total of 240bn euros.