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.
But sources said Seehofer had not yet submitted his resignation, adding that some in the party were attempting to change his mind over the decision.
The dispute over how to handle migration has plunged Merkel's 3-month-old "grand coalition" into crisis and weakened the conservative leader, who has served for almost 13 years, forcing German officials to scramble to try to reach deals with individual European Union members.
Merkel maintains that a plan to regulate immigration that European Union leaders approved Friday and bilateral agreements in principle that she hashed out with some countries for them to take back migrants would accomplish what Seehofer seeks. The CSU is facing a state election in Bavaria in October, and has opted to take a hard-line anti-immigration stance as its main campaign strategy.
If Seehofer goes ahead and resigns, the move would leave Merkel once again fighting for her political life.
When Merkel rejected the plan two weeks ago, Seehofer threatened to unilaterally implement his border control measures, using his authority as interior minister over the federal police.
Speaking outside the CSU headquarters after a leadership meeting, Seehofer told reporters "I said yes, that I would make both posts available, which I will do within three days". The German leader would not comment Sunday on the outcome of the talks.
This was a direct contradiction of Merkel's account of the meeting, which she delivered in an interview with public broadcaster ZDF on Sunday. "Now that the CSU is behaving more aggressively towards Merkel and the CDU, those normally critical have come to her support".
The CSU's Bavarian state premier Markus Soeder said at a political meeting on Saturday: "Of course what has been achieved in Brussels is more than we originally thought".
Deputy CDU leader Armin Laschet said the party's position is "independent of Horst Seehofer or Angela Merkel, because we want the European solution".
"I'd like the CDU and CSU to continue working together because we're a success story for Germany", she said.
Will divisions over migration split Germany's conservative coalition and spell the end of the line for Chancellor Angela Merkel?
That would likely end the seven-decade partnership of the CDU and CSU, which have a joint parliamentary group, in turn leaving the government just short of a majority.
Merkel has insisted that the migration crisis - which has seen tens of thousands of North African and Middle Eastern asylum seekers enter European Union countries in recent years - is a matter for Europe as a whole to resolve.
Mr Seehofer said the deal was affecting the "credibility" of his role as party leader and warned he saw "no alternative" but to turn back some migrants at the German border.