Here are the three main stumbling blocks on which they will have to reach common positions to avoid snap elections.
Migrants and refugees has been Germany's toughest political issue since a mass influx from 2015 brought some 1.2 million asylum seekers, sparking a backlash that has seen the far-right AfD party enter parliament.
Merkel's CDU and especially their more conservative CSU allies from Bavaria, which saw tens of thousands of refugees cross the border from Austria, are pushing to limit Germany's annual intake to a benchmark figure of 200,000.
The Greens, who have long promoted migrant rights and a multicultural society, reject such a limit and are pushing for a resumption of family reunions for those who have been granted temporary refuge in Germany.
They also reject a demand from the other parties to declare the North African nations of Algeria, Morocco and Tunisia “safe countries of origin”, which would raise the bar for asylum applications for their citizens.
Germany, a green energy pioneer long praised for its shift to renewables, now risks missing its carbon reduction targets, largely because coal plants still account for some 40 percent of electricity production.
All parties say they are committed to Germany's ambitious carbon goals for 2020 and beyond, but are fighting over how to do it.
The Greens have backed off their campaign demands to phase out coal as well as combustion engine vehicles by 2030, in the face of tough opposition from the other parties which worry about the impact on the energy and auto sectors and job losses.
But they have reportedly rejected an offer to close Germany's 10 dirtiest coal plants, having originally demanded that 20 be shuttered.
They are also pushing for a shift toward organic agriculture and steps to protect biodiversity and boost animal welfare.
All parties are pro-European and in favour of reforming the bloc, but divided on how, especially when it comes to paying for it.
While the Greens are sympathetic to the vision of French President Emmanuel Macron and his proposals for a common eurozone budget and finance minister, the CDU is sceptical and the FDP hostile.
The FDP opposes any measures that would lead to the pooling of debt or the transfer of German cash to troubled economies.
The Greens are more open to the notion of “solidarity” with eurozone countries in trouble.
On Turkey, with whom Germany is embroiled in a bitter dispute centred on civil rights, the CSU wants to push for an end to Ankara's long-stalled EU entry talks.
The Greens argue that the door should be kept open for the era after President Recep Tayyip Erdogan leaves office.