During the course of his press conference, Mr Seehofer said he had already presented a 63-point "master plan" to Mrs Merkel.
At issue Monday was whether the CSU would give her time to try to reach such a deal.
Mrs Merkel says she will report back on July 1.
As the German chancellor met with leaders of her Christian Democrats (CDU) on Sunday in an attempt to divert the collapse of her fledgling administration, Seehofer emerged from emergency talks with his Christian Social Union (CSU) saying he had no intention of toppling Merkel.
With an eye on October's Bavaria state election, the CSU is anxious to assure voters that it has a roadmap to curb the migrant influx. His party holds a leadership meeting Monday which could authorize Seehofer to push through his demand.
Seehofer's "migration master plan", which Merkel last week refused to endorse, would see asylum seekers arriving at Germany's borders turned away if they have no identification papers, have already had an asylum claim rejected in Germany, or are already registered in another country in the EU - proposals that rights group say contravene European and worldwide agreements.
In that scenario, Merkel could lead a minority government with the third party in the coalition, the Social Democrats (SPD), or call fresh elections that would likely benefit only the AfD, he said. Migration is "a European challenge that requires a European answer", Merkel said in her weekly podcast released Saturday.
That "would be the end of the government and the alliance between CDU and CSU", an unnamed CDU source told Bild.
'We want a solution for sending back refugees at our borders, ' he said. But he said the aim should be a "consensual solution" and wrote that it was "of decisive significance that the European Union summit at the end of June finally makes decisions that recognize Germany's burdens in migrant policy".
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"Angela Merkel's coalition is coming under increasing strain over the migrant issue".
"The situation is serious but still solvable", he wrote.
But the rest of the European Union is now engaged in an unseemly squabble over refugees as the new anti-immigrant Italian government has drawn the ire of Prime Minister Macron of France by refusing docking rights to a refugee ship.
Austria, governed by a center-right and far-right coalition, has also vowed to take a much tougher stand.
She has pledged instead to seek bilateral agreements with arrival and transit countries and a wider solution by the European Union, which holds its next summit on June 28-29, while admitting that this plan is "ambitious". But pointedly, Kurz announced the plan with Seehofer, not Merkel, standing by his side.
Germany had the most asylum seekers of any European nation, with more than 200,000.
It wasn't immediately clear what she might offer other countries in talks.
"There is a context here of rising criminality, which to a statistical degree is part of the migrant problem", he said, adding that Democrats in the USA and Merkel in Germany both "neglected to their peril" the opposition among citizens to illegal immigration and lax border security.
If Seehofer decides to go it alone, Merkel would have to fire him, sparking an unprecedented CDU-CSU split, political scientist Heinrich Oberreuter told business daily Handelsblatt.