Last week he threatened to turn asylum seekers away from Germany's borders unless Mrs Merkel reached an acceptable deal with other European Union partners.
In Berlin, Chancellor Angela Merkel gathered with senior members of her Christian Democratic Union (CDU) to discuss the latest developments.
Shortly after the reports of Seehofer's possible resignation, CDU General Secretary Annegret Kramp-Karrenbauer appeared before reporters to reaffirm the party's commitment to Merkel's asylum policy.
In a document sent to leaders of her coalition partners, Chancellor Angela Merkel listed 14 countries which had agreed to take back migrants.
The CSU had pressured Merkel for weeks to either agree to its plan to turn away at the border asylum seekers already registered in other countries or to achieve European Union deals that would have the same effect, setting her a deadline of Sunday July 1.
Leaders and MPs from the Bavarian party will meet in Munich to decide their response, while Merkel and her top lieutenants will gather in Berlin.
In a televised interview on Sunday, Merkel said "The sum of everything we have decided has the same effect [as national measures]".
The Social Democrats said Monday there was no guarantee they would back whatever compromise the CSU and CDU might find.
"Seehofer, who has nearly totally lost power in the CSU, now appears to want to bring down the chancellor in a kind of indirect political suicide. ending his career under the motto: first me, then the party, and maybe then the country", wrote the Hannoversche Allgemeine newspaper in an editorial.More news: Spurs and Lakers believe Kawhi Leonard trade clinches LeBron to LA
In the meantime, Bavaria state premier Markus Soeder said that Bavaria does not want to risk the stability of the German government, adding he hoped for a compromise with Merkel in the migrant policy row. The cantankerous Bavarian leader, who turns 69 this Wednesday, coyly said he would make a final decision on his job within three days.
On Sunday, Ms. Merkel's party passed a resolution backing her stance.
Mr Seehofer, whose party faces a state election in the autumn with a strong challenge from the far-right Alternative for Germany party, has threatened to turn away at the borders migrants whose asylum requests Germany already rejected or who already sought asylum elsewhere in Europe.
If Mr Seehofer were to go ahead with his policies, the dispute could end the decades-old conservative alliance between the CSU and the CDU.
He said: "After intensive discussions between the CDU and CSU we have reached an agreement on how we can in future prevent illegal immigration on the border between Germany and Austria".
Last November, when Merkel was negotiating her fourth coalition, the long-time chancellor told German broadcaster ZDF she was "very sceptical" about leading a minority government and would rather call new elections.
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.
Dealers said the clear win might settle one source of political uncertainty, but Obrador was also expected to sharpen Mexican divisions with U.S. President Donald Trump.
Since then, more than one million people have arrived in Germany, while Merkel's governments have repeatedly tightened immigration and asylum laws.