The moderate Democratic Left party in Greece says it will not join pro-bailout parties in a coalition without the more radical far left Syriza.
The Greek president has called the four main parties, including the centre-right New Democracy and the Socialist Pasok, to try to form an emergency government to avoid new elections.
But Syriza said it would not attend because it could not back any coalition which supported austerity.
A majority voted against last week.
"No unity government can emerge," Fotis Kouvelis, head of the Democratic Left party, told Greek television.
"A government without Syriza would not have the necessary popular and parliamentary backing," said Mr Kouvelis. Continue reading the main story Analysis image of Mark Lowen Mark Lowen BBC News, Athens
There is still a very slim possibility that there could be success. Democratic Left's leader Fotis Kouvelis has said he will still go to the talks this evening at the presidential palace with the two other big parties.
So that will really be the final throw of the dice.
If at the 59th minute of the 11th hour he says "Yes", Greece will be able to cobble together some kind of albeit very shaky coalition. If he says "No", this country will take a leap into the darkness, and face fresh elections within four weeks.
According to opinion polls, fresh elections could usher in a government that wants to turn its back on the bailout and on austerity, and therefore from the EU's point of view, possibly turn its back on the euro.
EU finance ministers are due to meet in Brussels to discuss the Greek crisis later on Monday.
The fear over holding new elections is that parties that oppose austerity measures that are a condition of Greece's bailout deal might do well again in new polls, says the BBC's Gavin Hewitt in Brussels.
And with no sign Europe's leaders are prepared to renegotiate the deal, Greece could end up leaving the eurozone.
For the first time, some central bankers have spoken openly about the consequences of a Greek exit from the single currency, our correspondent adds.
President Papoulias had invited four parties, including Syriza, to further talks.
Both New Democracy and Pasok have so far been unable to form a new coalition. Continue reading the main story Greece in trouble
17 May: State opening of parliament; deadline for formation of new government 6 May: Election held; voters desert main parties and back anti-austerity groups; no party in overall control so coalition needed 12 Mar 2012: Second bailout signed off involving 130bn-euro loan and massive debt write-off 10 Nov 2011: Technocrat PM Lucas Papademos installed after social unrest and economic chaos bring down Pasok government 2 May 2010: EU/IMF agree 110bn-euro bailout in return for austerity measures after Greek debt downgraded to junk status
Q&A: Greek debt crisis