Her hardline interior minister Horst Seehofer on Monday gave her a two-week ultimatum to reach such deals, threatening he would otherwise defy her and shutter German borders to these migrants, a move that would likely end their coalition.
"We do not really have a handle on this whole issue of migration", Seehofer told reporters in Munich as Merkel spoke in Berlin.
In Brussels, the European Union asylum office said Monday the number of people applying for worldwide protection in Europe plunged past year but remains higher than before 2015, when more than one million migrants entered, many fleeing the war in Syria.
Her interior minister, Horst Seehofer, has been calling for Germany to turn back migrants at the border who have previously applied for asylum or registered as asylum-seekers in other European countries.
The CSU exists only in Bavaria, while Merkel's less conservative CDU is in Germany's other 15 states.
Merkel therefore needs to balance Seehofer's demands to turn away migrants who have been registered in another European Union country - often Italy or Greece as their first port of call - with the frontline nations' call for solidarity.
Merkel's liberal approach to immigration has been controversial in Germany, however, particularly on the right.
Several high profile crimes by migrants have also fuelled public anger. And the issue has been inflamed recently by the rape and murder of a 14-year-old German girl that authorities believe was carried out by an Iraqi asylum seeker.
With an eye on October's Bavaria state election, the CSU is anxious to assure voters that it has a roadmap to curb the migrant influx.
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She also warned the minister, Horst Seehofer , against defying her, saying she is ultimately in charge of government policies. At issue Monday was whether the CSU would give her time to try to reach such a deal.
"France and Germany have agreed to set up a Eurozone budget. These are all questions that we will discuss in the coming months and where we want to work very closely together", Merkel said.
Merkel, the EU's longest-serving leader, welcomed Monday's compromise in a dispute that has threatened to unravel her new coalition, and said her Christian Democrat (CDU) party would decide how to proceed after the two-week deadline elapsed.
France's President Emmanuel Macron, center left, and German Chancellor Angela Merkel, center right, pose with their governments after a meeting in the guest house of the German government in Meseberg north of Berlin, Germany, Tuesday, June 19, 2018.
Merkel droned out the usual platitudes about lessons being learned etc, and that Jewish people living in Germany should not have to suffer like this again.
Central and eastern European Union nations such as Hungary and Poland have either refused outright or resisted taking in refugees under an European Union quota system, and Austria has also taken an uncompromising stance.
"We are opening a new chapter", Merkel said after the talks.
Merkel is to meet with her party's leadership July 1 to discuss the outcome of meetings with European Union leaders to be held in Brussels on June 28 and 29.
She said yesterday that she supports 62 of his 63 points.
For the the Bavarian party, however, "the real dilemma they face is between accepting the bilateral deals Merkel brokers, or enforcing their plan unilaterally with the risk of Seehofer being fired".
Perhaps reflecting that nervousness, Seehofer later added: "Nobody wants the end of the coalition, a collapse of the alliance or the fall of the government".