The rebel minister is leader of the Christian Social Union in Bavaria, whose parent party is Mrs Merkel's Christian Democratic Union (CDU). The CSU lost thousands of voters to the far-right Alternative for Germany (AfD) party in federal elections a year ago, and is fearful of further losses.
A CSU leadership meeting Monday in Munich unanimously backed Seehofer's plan to give Merkel until the end of the month to find a solution with other European Union countries.
Perhaps reflecting that nervousness, Seehofer later added: "Nobody wants the end of the coalition, a collapse of the alliance or the fall of the government".
Interior Minister Horst Seehofer had released new crime figures in May that pointed to an overall decline in Germany over the past year.
"The people of Germany are turning against their leadership as migration is rocking the already tenuous Berlin coalition", he wrote.
Rightwingers in her delicate governing coalition - as well as several northern European countries - remain hostile to France's push for a common eurozone budget, fearing taxpayers could foot the bill for fiscal irresponsibility in southern EU states.
Merkel said she will hold talks on bilateral agreements with other European countries at and around a June 28-29 EU summit.
There is, she insisted, "nothing automatic" about what happens next.
Merkel quickly made clear that she disagreed. "And I view this issue as decisive for keeping Europe together", she said.
The tweets come as Merkel is in a stand-off with her interior minister over new immigration curbs. The document, which has not yet been revealed to the public, reportedly envisages turning back certain categories of asylum seekers right at the German border as well as reduction of benefits for asylum seekers living in Germany.
"We wish the chancellor success in this", he said. "We finally want to have a solution for the return of refugees at our borders which is fit for the future", he added.
He said he had only recently learnt that migrants who had already been denied asylum by Germany or been issued re-entry bans were still being allowed back in.
The spat over immigration has laid bare the deep tensions in a fractious German government that took office only in March, after almost six months of postelection haggling, and exposed the limits of Merkel's authority.
He has the nuclear option of seeking approval from his party to shut Germany's borders immediately in defiance of Merkel, or the less aggressive choice of giving her an ultimatum of two weeks to sort out a deal with other European Union nations.
Merkel and Seehofer papered over the cracks ahead of last year's national election, but support for both parties still dropped significantly.
Seehofer wants Germany to have the right to reject migrants who have already registered in another European Union state but Merkel opposes any unilateral move by him that would reverse her 2015 open-door policy and undermine her authority.
Merkel, who opened the country to more than a million people fleeing war, oppression and poverty in 2015 and 2016, vehemently opposes such a move and is pressing for a Europe-wide solution to the continent's continued struggles with migration. Around 30 per cent came from conflict-torn countries such as Syria and Iraq.