Seehofer and his party met Sunday seeking a response to a hard-fought agreement to reduce migration into the European Union and so-called "secondary migration" between member states hammered out by Merkel at a leaders' summit last week.
Top CSU officials say they want to stay in government with Mrs Merkel's CDU.
Mr Seehofer complained that he had held a "conversation with no effect" with Mrs Merkel on Saturday.
Interior Minister Horst Seehofer, who heads the Bavarian-only CSU, wants Germany to turn away some asylum-seekers at the country's borders, but the chancellor has insisted on Europe-wide solutions.
Merkel will try to set the tone an hour earlier in an interview with national television, before chairing an executive-board meeting of her Christian Democratic Union in Berlin.
The sources said that in Sunday's private party meeting, Seehofer said he had a "conversation with no effect" with Merkel when they held talks on Saturday about the European Union migration deal, AFP news agency reported.
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".
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Germany's CDU chairwoman Annegret Kramp-Karrenbaur said Ms Merkels party now have a deal that will prevent migrants registered elsewhere from entering Germany.
Seehofer, 68, has been a headache for Merkel, especially over her open-door migrant policy, and the last three years have been punctuated by a series of showdowns between the two.
Mr. Seehofer, after talks with Ms. Merkel on Saturday evening, complained the conversation was "ineffective" as the chancellor refused to budge her position on the European Union deal.
"And then Chancellor Merkel is entitled to dismiss her minister of interior".
CSU leaders are divided over how to face down a challenge from the anti-immigration Alternative for Germany (AfD) in October's regional election.
Ending the alliance with the Christian Democrats (CDU) would leave the government without a parliamentary majority.
She reiterated her determination to act in a way that was "not unilateral and that are not to the detriment of third parties" - highlighting her continued opposition to Seehofer's demand to turn back refugees at the border.
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If Seehofer does quit, the CSU could offer a replacement interior minister if it aims to remain tied to Merkel's CDU party.
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.
The political crisis is one more sign of a growing rift in Europe between those who want to work together to reduce the number of migrants and refugees entering the bloc, and those who have grown exhausted of failed attempts to reach bloc-wide agreements and are taking unilateral action.
Volker Bouffier, premier of the state of Hessen and Merkel's close ally, said that the CSU should be pleased at what it had achieved so far as he arrived at a similarly protracted meeting of his and Merkel's CDU in Berlin.
Bavaria's CSU Prime Minister Markus Söder said "we're ready for compromises" and "for us now there is no exit from the government".
Its success was based on strident anti-immigration rhetoric, attacking Mrs Merkel's decision to let in more than a million asylum seekers in 2015-2016.
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