Interior Minister, and CSU chairman, Horst Seehofer wants to see Germany refuse entry to migrants who have already been registered in another European Union country such as Italy or Greece, where they often land from the Middle East or North Africa.
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.
Merkel's junior coalition partner - the Bavarian Christian Social Union (CSU) - is under pressure from the nationalist Alternative for Germany (AfD) party, which became Germany's largest opposition party in the elections a few months ago. His party holds a leadership meeting Monday which could authorize Seehofer to push through his demand. That banished - if only for now - the spectre of Seehofer pushing through his proposal in defiance of the chancellor, which would risk bringing down her government.
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.
Seehofer's ultimatum has become one of the biggest challenges to Merkel's authority since she took power almost 13 years ago.
In that scenario, Merkel could lead a minority government with the third party in the coalition, the Social Democrats (SPD), or call fresh elections that would likely benefit only the AfD, he said.
Bavarian governor Markus Soeder and the party's top federal lawmaker, Alexander Dobrindt, have been even more vehement than Seehofer in demanding immediate action on migration.
"The overwhelming majority of the German population supports this idea" - of turning the previously registered back at the borders - "and that's why we want to provide support today for implementing it", he said.
Seehofer and Merkel have long had an awkward relationship.
Sri Lanka skipper Dinesh Chandimal charged with ball-tampering by ICC
The charges had been laid by on-field umpires Aleem Dar and Ian Gould , and third umpire Richard Kettleborough on Saturday. In March this year, Australia were caught tampering with the ball in the third Test against South Africa in Cape Town.
The German chancellor wants the European Union to give states more power to turn away refugees to stop Germany being the only one with the policy.
But Merkel says that would leave countries at the EU's southern periphery alone to deal with the migrant influx.
"This is a European challenge that also needs a European solution". "I see it as one of the most decisive issues in holding Europe together".
Mike Mohring, the CDU leader in the eastern state of Thuringia, told the daily Die Welt that he expects the CSU to give Merkel two weeks to find a European solution.
"Angela Merkel's coalition is coming under increasing strain over the migrant issue".
In a column for in the Frankfurter Allgemeine Zeitung, meanwhile, Seehofer wrote: "It is essential that the [European Union] summit takes a decision at the end of June".
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.
Failure to reach a deal that can be presented as a breakthrough at the European level could precipitate a full-blown crisis in Germany that might topple Merkel after nearly 13 years as chancellor.
She has pledged instead to seek bilateral agreements with arrival and transit countries and a wider solution by the European Union, which holds its next summit on June 28-29, while admitting that this plan is "ambitious".