With both the left- and right-wing blocs struggling to build a majority, speaker Andreas Norlen's office said he would continue informal talks with party leaders before providing further information on the process "this week".
He could give the nod to outgoing prime minister Ulf Kristersson, who heads a caretaker government after narrowly losing the September 13 election.
But he could also tap the leader of the centre-left Social Democrats, Magdalena Andersson, Sweden's first woman prime minister in 2021-2022.
Norlen tasked her with trying to form a government after the election, in which her left-wing bloc won 176 seats -- three more than Kristersson's right-wing bloc which includes the far-right Sweden Democrats.
But she threw in the towel on Monday, after failing to unite the three other parties in her bloc due to deep ideological differences.
Kristersson, 62, would also be hard pressed to secure a parliamentary majority.
He said he was ready to explore options to find a majority among "non-socialist" parties.
Kristersson is seen as wanting to court the Centre Party, which is now part of the left-wing bloc but used to belong to the right wing.
Centre Party leader Elisabeth Thand Ringqvist has however refused to support any government that gives any influence to far left or far right parties.
Kristersson, in power since 2022 thanks to the unprecedented support of the far right, had vowed the anti-immigration party would be given cabinet posts for the first time if the right wing won the election, though he has backtracked since.
The parliament speaker has four chances to put a candidate prime minister to a vote in parliament before a new election has to be called.
Sweden has become accustomed to lengthy negotiations to form a government. After the 2022 election it took 37 days and 134 days in 2018.
Most observers have predicted that Kristersson would likely fail if tasked with forming a government, and the speaker would then solicit Andersson again, who would ultimately manage to unite her bloc under pressure to avoid a new election.