SPD's leader Martin Schulz said last month that election result meant voters had rejected the option of another "grand coalition" between the SPD and the CDU/CSU, and gave Social Democrats the role of main opposition.
The setbacks Merkel's bloc suffered in the September election were in part due to the rise of the far-right, anti-immigration AfD which took millions of votes from all mainstream parties.
Schulz had said on Monday that, as he interpreted the election outcome as a vote against forming another grand coalition, he was unwilling to enter talks of reviving the deal with the CDU but also said that the SDP was ready to go into a new round of elections.
However, Merkel's talks with two other parties, the left-leaning Greens and pro-business FDP, collapsed early this week when the FDP unilaterally pulled out.
But he is under pressure to reconsider an alliance amid the political impasse.
"After meeting this week with the leaders of the CDU, CSU and SPD the federal president has agreed to meet a joint meeting at Schloss Bellevue", Steinmeier's spokeswoman said in a statement.
Following a night of debate between the party's leaders, SPD Secretary General Hubertus Heil told reporters in Berlin: "The SPD firmly believes that there must be talks".
Schulz' leadership could potentially be called into question at the party's conference in December in an attempt to rebuild from its poor showing at the election, with unnamed SPD figures telling German outlet Süddeutsche Zeitung that he would unlikely be the party's candidate were another election called.
Backing for a new government could come in various forms other than a coalition, including a formal agreement not to obstruct a Merkel-led minority government or an informal agreement to tolerate it.
The turnabout by the centre-left party, which had said it would go into opposition after suffering its worst result in 70 years in a September election, could help avert a disruptive repeat election in Europe's economic and political powerhouse.