The head of Greece's conservative party, Antonis Samaras, says he is unable to form a coalition government.
Samaras, whose New Democracy party won the most votes in Sunday's election, was given the first chance to form an administration by President Karolos Papoulias.
But after holding talks Monday, Samaras failed to convince other parties to join his bloc so he could gain the majority in the 300-seat parliament.
Now, the leftist party that came in second will get a chance to cobble together a ruling alliance. If those talks fail, the party that came in third will get the mandate.
No agreement could force new elections next month.
Meanwhile, nervous European leaders maintained pressure on Greece to continue respecting the terms of a financial bailout. In casting their ballots, Greek voters appeared to favor parties that opposed austerity requirements imposed on Athens in return for more than $300 billion in two bailouts.
Also Monday, German Chancellor Angela Merkel said it was "of utmost importance" that Greece sticks to its reform path, although she acknowledged that this was "difficult."
A spokeswoman for the European Commission said Brussels "hopes and expects that the future government of Greece will respect the engagements that Greece has entered into."
The bailout deal, struck in February with the European Union, the International Monetary Fund and the European Central Bank, aims to clear the way for Greece to return to financial markets by 2015.
Some information for this report was provided by AP and Reuters.