The leader of Greece's main socialist party, Evangelos Venizelos, is due to meet conservative Antonis Samaras in an attempt to form a unity government.
Mr Venizelos is the third leader to try to form a coalition after Sunday's election produced a hung parliament.
The socialist Pasok and centre-right New Democracy parties governed in coalition until last Sunday's election.
Voters angry at austerity measures abandoned them for parties rejecting the terms of an international bailout.
Analysts say Friday's talks are unlikely to succeed. If they fail, the president must ask all political leaders to make one final effort to form a government before calling fresh elections.
Pasok dominated Greek politics for most of the past four decades, but saw its support slashed on Sunday - coming third with just 41 seats in the 300-seat parliament.
Mr Samaras's New Democracy won the most votes, taking 108 seats, but he was unable to woo other party leaders.
Left-wing coalition Syriza was the second biggest party, but its leader Alexis Tsipras also failed to form a government because of his insistence on rejecting austerity measures demanded by the EU and IMF.
Pasok and New Democracy between them have 149 seats, two short of a majority.
However, on Thursday Mr Venizelos said he had made progress after meeting the leader of the Democratic Left party, which has 19 seats.
Its leader, Fotis Kouvelis, said he was willing to join a broad-based government that would keep the country in the euro but disengage it from the bailout.
"There is a very slim chance for a coalition if Kouvelis agrees," one socialist party official quoted by Reuters said. "But his party is split right down the middle."
The BBC's Mark Lowen in Athens says Pasok is deeply unpopular - seen as the architects of austerity, and tainted with allegations of corruption.
As the prospect of fresh elections loomed, an opinion poll published late on Thursday put Syriza in first place with nearly 28% of the vote - up from 16.8 - winning 128 seats.
The Marc survey for private Alpha TV put New Democracy in second place with 20.3% and 57 seats, and Pasok third with 12.6% of the vote and 36 seats.
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.
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.
Both Germany and the EU have made clear they expect Athens to honour its commitments.