Antonis Samaras' New Democracy party came first in Sunday's vote, winning 129 of Parliament's 300 seats - not enough for him to govern alone. He is seeking an alliance with the third-placed Socialist PASOK and the smaller Democratic Left party. New Democracy and PASOK took part in a 6-month coalition government earlier this year.
PASOK leader Evangelos Venizelos met Tuesday with Democratic Left head Fotis Kouvelis, who told reporters afterward that a deal could potentially be achieved "within hours."
Kouvelis said parties first had to agree on a policy platform, and on who would be appointed to the cabinet.
"The country must have a government," he said.
Venizelos, who has pressed for an agreement by Tuesday night, said he was confident a deal could be reached.
"I am confident after my meeting with Mr Kouvelis, our views are very close," he said.
Both the conservatives and PASOK have pledged to respect the commitments for further austerity and reforms that Greece undertook as conditions for two massive international bailouts. The Democratic Left is anti-austerity in principle, but wavering.
The second-placed, anti-bailout Syriza radical left party has refused to join in a coalition. In order to pass most laws, a government needs a simple majority of 151 seats. But to have any chance of success, it would require considerably more than that, especially amid the social turmoil of the past two-and-a-half years of harsh austerity in Greece.
Together with PASOK and the Democratic Left, Samaras' party would have 179 legislators, enjoying a strong majority of 28 seats.
Samaras' three-day mandate to form a government expires on Wednesday. If it fails, Syriza would get a chance to form a government, followed by the Socialists. But Syriza leader Alexis Tsipras has said he will not even try to create a government. Venizelos has advocated circumventing the mandate process and proceeding straight to a meeting of party leaders under President Karolos Papoulias.