SPD leader Olaf Scholz said he had a clear mandate to form a government, while his conservative rival Armin Laschet remains determined to fight on.
The two parties have governed together for years.
But Mr Scholz says it is time for a new coalition with the Greens and liberals.
Preliminary results gave his party a narrow election win over the conservatives who suffered their worst-ever performance.
Despite this, Mr Laschet said his party had given him its backing to enter talks with coalition partners, pushing Germany towards a potentially protracted power struggle.
The Greens and pro-business Free Democratic Party (FDP) attracted the most support from the under-30s, in an election dominated by climate change and by differing proposals on how to tackle it.
The Greens made history with almost 15% of the vote, even though it was well short of their ambitions.