The minister, Horst Seehofer of the Christian Social Union (CSU), wants police to have the power to turn away undocumented migrants at the border.
Mrs Merkel has held emergency talks with her Christian Democrat (CDU) MPs.
She wants a new deal at EU level over migrants. She was widely criticised for letting in about one million in 2015.
Mr Seehofer has been a long-standing critic of that open-door policy, which was summed up at the time by Mrs Merkel's phrase "we can manage".
He says German border police should turn back all asylum seekers who lack IDs and those already registered in another EU country.
Mrs Merkel sees that as a "go-it-alone" policy which would further hurt Italy and Greece, countries whose capacity to house migrants is stretched to breaking point. Among them are many refugees from the Syria war and other conflict zones.
While Mrs Merkel's CDU MPs gave her their support on Thursday, their Bavarian colleagues from the CSU decisively backed their leader, Mr Seehofer.
The "sister" conservative parties have been in alliance at the federal level since 1949.
There is a risk that the CSU could break away, or at least trigger a confidence vote in Mrs Merkel's leadership. In the 709-seat Bundestag (lower house) the CDU has 200 seats and the CSU 46.
The ruling coalition with the Social Democrats (SPD) has 399 seats, but without the CSU that would fall to 353 - less than a majority.
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