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.
Merkel, who opened the country to more than a million people fleeing war, oppression and poverty in 2015 and 2016, vehemently opposes such a move and is pressing for a Europe-wide solution to the continent's continued struggles with migration.
German Interior Minister and leader of the Christian Social Union Party Horst Seehofer arrives at his party's headquarters in Munich on Monday, to attend a party leadership meeting to discuss the next steps in a deepening conflict with its sister party, the German Chancellor's CDU, centred on toughening immigration rules.
"No one in the CSU is interested in bringing the chancellor down, or dissolving the CDU/CSU parliamentary partnership or destroying the coalition", Seehofer told the Bild am Sonntag newspaper, adding that he did not want the asylum row to endanger the coalition government which is less than 100 days old.
As the two states refused to help, some 630 desperate migrants were left terrified and in desperate need of medical attention off the coast of Italy.
However it ends, the spat has laid bare the limits of Merkel's authority in a fractious government that took office in March after almost six months of postelection haggling.
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Austria, governed by a center-right and far-right coalition, has also vowed to take a much tougher stand.
"We just want to finally have a sustainable solution to send refugees back to the borders".
Merkel and Seehofer papered over the cracks ahead of last year's national election, but support for both parties still dropped significantly.
Merkel also said Germany and France would try to give new momentum to EU cooperation on foreign, defence and security policy and on further developing the economic and currency union. "I think this is issue is one of the most decisive for the cohesion of Europe".
If Merkel is given a two-week ultimatum, she would still face the challenge of persuading European Union governments to sign up to a common plan on the migrants. Seehofer has said these migrants should be turned away at the German border whereas Merkel has said this can only happen with the agreement of the relevant European Union states.
Merkel already has meetings scheduled Monday with Italian Premier Giuseppe Conte - the head of a new, populist government whose interior minister has pledged to deport tens of thousands of migrants - and Tuesday with French President Emmanuel Macron. Merkel believes it is "in Germany's interest" to reach an agreement on managing migration "in close partnership with our European neighbors".
The German government said no special EU summit is planned, and that would be a matter for EU leaders in Brussels, but "of course the German government is holding talks with various member states and the (EU) Commission" about immigration in Europe.