She insists that unilateral action by Germany would trigger a wider crisis, and will now use the run-up to a June 28-29 summit of European Union leaders to try and forge some sort of agreement.
The Christian Democratic Union (CDU) politician said it was still unclear in Berlin what the essence of the disagreement between the Chancellor and Mr Seehofer was and that other members of the coalition had been left "in the blue".
The CSU appeared to be at pains at the weekend to limit the damage done to the conservative alliance amid growing evidence the German government could be about to collapse.
Horst Seehofer, the CSU interior minister, whose party is facing regional elections in October, is calling for Germany to turn away refugees who have also registered in other European countries. He favors stronger screening of refugees' possible terrorist connections and speeding up deportations of failed asylum seekers, according to Focus Online.
Merkel's opposition to the proposal reportedly angered Seehofer.
However, a spokesman said Mrs Merkel is in close contact with those states.
The minister added that he hoped an agreement could be finalised on Tuesday, when German Chancellor Angela Merkel and French President Emmanuel Macron are scheduled to meet near Berlin.
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.
The BAMF has been at the heart of a scandal around the right of asylum for several weeks, with one of its branches in Bremen, northern Germany, under investigation for allegedly favouring more than 1,200 asylum applications.
The CSU's Bavarian state premier Markus Soeder said that the ongoing "asylum tourism" meant that, as a government, "we're nearing the endgame when it comes to our credibility".
"The people are running out of patience", he told Bild daily.
Because of its troubled history and Merkel's welcoming stance in 2015, Germany seemed somewhat more immune than other countries to the populist wave washing over Europe.
For Merkel, in power for over 12 years, the stakes could not be higher as she leads an uneasy coalition government with a narrow majority that took half a year to cobble together.
If he actually does so unilaterally in defiance of the chancellor, many observers believe Merkel would likely have to fire him - which in turn could effectively end her current governing coalition and the conservative parties' decades-old alliance in national politics.
Seehofer's party, one of three in Merkel's coalition, is poised to hand Merkel a two-week deadline to get a European deal facilitating the return of migrants to countries in which they were first registered. Merkel has said she is "favourable" to a French call to create a European Intervention Initiative of forces which could be deployed rapidly to deal with crises outside the framework of North Atlantic Treaty Organisation.
The fake news was quickly debunked as originating from satirical magazine Titanic, but not before it had been picked up by some major media, causing immediate dips in the euro and stock market.