Sunday's vote—the second national election in six weeks—again left no party with enough votes to form a government on its own. Antonis Samaras' conservative New Democracy party won the most seats in parliament and was leading efforts to forge a coalition.
The socialist PASOK party, led by former finance minister Evangelos Venizelos, came in third. But its 33 seats in the 300-member Parliament means it can form a government with New Democracy, which gained 129 seats. A coalition would have to have a minimum of 151 seats combined in order to form a government.
Both PASOK and New Democracy have said they will stick to Greece's international bailout commitments, although they want to renegotiate some of the harsh austerity measures imposed in return for the international rescue loans that have kept the country afloat since May 2010.
The election results eased concern that Greece faced an imminent exit from Europe's joint currency. A Greek exit from the 17-nation eurozone would have potentially catastrophic consequences for other ailing European nations and hurt the United States and the entire global economy.
As head of the party that came first, Samaras was given the mandate Monday to seek coalition partners. He has three days to reach an agreement, and if he fails the second party is given another three days to try. The radical left-wing anti-bailout Syriza party came in second.
"With Mr. Venizlos we agreed that within the deadline of my mandate ... a government of national salvation must absolutely have been formed," Samaras said after talks with the socialist leader. "We will of course have new meetings."
Samaras also met Monday evening with Fotis Kouvelis, the head of the small Democratic Left party that finished sixth in Sunday's vote. He said afterwards that the meeting was fruitful, and would be repeated Tuesday.
Kouvelis' party had been seen as a potential partner for PASOK and New Democracy after inconclusive elections on May 6. Those coalition talks collapsed after 10 days of wrangling, triggering last Sunday's elections.
Syriza has refused to join the other two parties in a government, saying it will not cooperate with any group that insists on implementing the harsh austerity measures taken in return for Greece's two international bailout agreements.
Venizelos, however, has insisted on a broad coalition. "The most crucial thing for us right now is to achieve the greatest possible range of consensus, and this must happen by tomorrow night at the latest," he said after his talks with Samaras.
He also criticized Syriza chief Alexis Tsipras for his refusal to join in governing Greece. "You can't have some people choosing the easy position of being in opposition and lying in wait for the government to fail—or rather trying to create the conditions for the government, that is the country, to fail," Venizelos said.
On the streets of Athens, the mood was mixed, with many saying party leaders must get their act together.
"The election result isn't strong enough to put people's minds at ease," said sandwich shop owner Mary Moutafidis, 57. "They still have to agree to form a government."
European Union leaders appeared relieved that a pro-austerity government had a good chance of being formed.
"Continued fiscal and structural reforms are Greece's best guarantee to overcome the current economic and social challenges," European Council President Herman Van Rompuy and European Commission President Jose Manuel Barroso said in a joint statement.
But an early market rally faded quickly Monday as investors turned their attention back to the other financially unstable economies in the Eurozone, Spain and Italy.
In Athens, stocks lost initial strong gains but still closed up 3.64 percent.
Greece has survived for more than two years on rescue loans from its European partners and the International Monetary Fund. The vital bailouts are conditional on the country continuing with its deeply unpopular package of spending cuts, and pushing through new structural reforms.
Athens has pledged to push through new savings worth nearly (EURO)15 billion ($18.9 billion), raise billions in company and real estate privatizations and sack about 150,000 civil servants.
Both New Democracy and PASOK want an extension of at least two years to the austerity and reform deadline, to alleviate pressure on a population exhausted by two-and-a-half years of deep income cuts and tax hikes.
Unemployment has soared to more than 22 percent, while Greece's economy is in a fifth year of deep recession.
Menelaos Hadjicostis and Elena Becatoros in Athens contributed.