News agency Dpa said that Horst Seehofer, the leader of the Bavarian-only Christian Social Union which is in coalition with Mrs Merkel, was planning to announce his resignation.
After further talks with a smaller group of CSU officials seeking to change his mind, Seehofer said he had agreed to meet again with Merkel's party before he made his decision final.
Seehofer had previously warned that, unless Merkel came up with a satisfactory EU-wide solution to dealing with migrants, he would use his power to make sure that asylum-seekers who applied for asylum in states other than those where they'd first entered, would be turned away at the border.
The CSU will face a state election in Bavaria in October and has opted to take a firm stance against immigration as part of its campaign.
Mr. Seehofer had already offered his resignation late on Sunday evening during the CSU leadership's meeting.
There were fresh signs on Sunday that the two leaders, entrenched in their positions, may fail to resolve their differences.
Crucially for Merkel, CDU lawmakers are backing her - so far.
Bavarian governor Markus Soeder arrives for a board meeting on the German migration policy in Munich, southern Germany, Sunday, July 1, 2018.
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To politically survive, Merkel could attempt a minority government, seek a new coalition partner in the ecologist Greens or pro-business Free Democrats, or orchestrate a no-confidence vote in parliament that could trigger new elections.
After an "ineffective" two-hour meeting on Saturday night between Mrs Merkel and Mr Seehofer, the two leaders met with their respective parties for separate meetings in Munich and Berlin on Sunday night.
At the national level, Merkel also proposed that migrants arriving in Germany who first registered in another European Union country should be placed in special "admissions centres" under restrictive conditions.
Merkel's government has been pushed to the brink over the migration issue after allowing more than one million asylum-seekers into Germany since 2015.
The EU and bilateral deals were "only possible because the chancellor enjoys respect and authority throughout Europe", Germany's EU Commissioner and CDU politician Guenther Oettinger said in an interview with the Frankfurter Allgemeine Sonntagszeitung weekly.
But the CSU and CDU together form a center-right force that has dominated national politics for decades.
Hans-Peter Friedrich, a member of the CSU leadership, said that Merkel's rejections of compromises submitted by Seehofer suggested she was happy to see him go.
EU leaders agreed at the summit to share out refugees on a voluntary basis and create "controlled centres" inside the European Union to process asylum requests.
Ahead of a hard Bavarian state election in October, the CSU is determined to show it is tough on migration.