Portuguese President Anibal Cavaco Silva is due to decide whether to name Socialist leader Antonio Costa as head of a coalition with far-left parties.
EU countries are eyeing developments in Lisbon nervously because of the far-left's anti-austerity policies.
Centre-right Social Democrat Pedro Passos Coelho won the 4 October election but lost his majority.
The president has consulted with political parties and has to decide who to nominate as prime minister.
Even if he chooses Mr Coelho, many believe he will still have to ask the left to form a government within weeks.
The Social Democrats could only command a minority government and would probably fall as soon as the opposition tabled a vote of no confidence.
As the political parties continue to battle over who will form the next coalition, there are fears that weeks of uncertainty could harm Portugal's economic recovery, more than a year after it exited the strict terms of its €78bn (£57bn) international bailout.
EU officials have threatened to take action against Portugal for missing a 15 October deadline to present its draft 2016 budget.
Portugal is still running one of the highest budget deficits in the eurozone.
Mr Coelho's Social Democrats have promised to present a budget, but the two left-wing parties campaigned strongly against his outgoing government's record of harsh austerity.
The Left Bloc is seen as allied to the anti-austerity Syriza party, which for months tried to renegotiate the terms of Greece's eurozone bailout.
Portugal's Communist Party is regarded as anti-euro and anti-Nato, although it is thought to have moderated its eurozone policies in recent weeks.
If Mr Costa's Socialists are eventually chosen to lead a left-wing coalition, it would be the first time since the fall of Portugal's dictatorship in 1974 that a right-wing president appointed a government backed by communists.