After the September 9th election left the two main blocs separated by just one seat, arriving at a compromise could be difficult, and the first round of talks last week didn't break the deadlock.
It is the job of newly-elected speaker Andreas Norlén to put forward a proposal for who should become prime minister, but this is only done after talks with the different leaders. On Friday, representatives from the parties appeared to call on Norlén to give one of the eight party leaders the go-ahead to start negotiations on forming a new government.
Both the centre-left bloc and the centre-right Alliance (Moderate Party, Christian Democrats, Centre Party and Liberal Party) believe they should be the one to lead Sweden's next government. If they can't find any common ground, the far-right Sweden Democrats, who are the third largest group, will be influential.
One possible result of Tuesday's talks is that Moderate Party leader Ulf Kristersson could be asked to try to form a new government, which would force Social Democrat leader Stefan Löfven to talk with the centre-right Alliance.
The Social Democrats' spokesperson has said the right person to lead negotiations is Löfven — who was ousted from his job after losing a confidence vote last week, but continues to lead a caretaker administration.
Another possibility is that the Moderates and Christian Democrats might try to form a government with the support of the far-right Sweden Democrats (SD) but without the Alliance's other two parties, the Centre and the Liberals, which have said they will not work with the SD.
Norlén has four attempts to get parliament to agree to a new prime minister, or at least convince enough MPs to abstain and not actively vote against the candidate.
If they fail to agree on any of the four proposals, a new election must be held within three months. However, this has never happened in Swedish history.