Horst Seehofer abruptly called off a press conference scheduled to announce his new asylum policy as it emerged Mrs Merkel had blocked the plans.
On Tuesday, some senior members of Merkel's Christian Democrats (CDU) opted to back her hard-line Interior Minister Horst Seehofer, a Bavarian conservative who has always been a thorn in the chancellor's side over migration.
"It is important to have a sustainable solution, which is why I'd like to discuss this on a European level", she told reporters at a joint news conference alongside her Austrian counterpart Sebastian Kurz.
Seehofer is now trying to pull Germany's new government to the right and to adopt a harder stance on migration to win back territory from the populist Alternative for Germany (AfD) party which has gained their first seats in the legislature.
Angela Merkel was facing a possible government crisis on Tuesday as a deep rift opened with one of her coalition partners over migrant policy.
Mr Kurz heads a right-wing government and has similar views to Mr Seehofer, he said: "We must decide who come to Europe, not the smugglers".
The anti-immigration AfD said Seehofer must prevail.
He has said he believes that his proposed measures are "necessary" to restore "order in Germany" and that he would not "publish a half-baked plan with lazy compromises".
Touching the thorny issue of Germany's relatively low defence spending, Merkel acknowledged that Trump's criticism was partly correct and that Berlin had to do more to reach NATO's goal of spending to 2 percent of economic output on defence.
Merkel leads a coalition in Berlin of her Christian Democrats, their Bavarian allies CSU to which Seehofer belongs, and the Social Democrats (SPD).
"Of course they must be turned back at the border", he said. "We make commitments and keep them".
It too rejects stepping up German border controls of asylum seekers, which it says goes against the spirit of the Schengen passport-free zone.
The Austrian Chancellor Sebastian Kurz landed in Berlin yesterday for talks on European Union migration policy with Chancellor Merkel. "And then hope that the European Union will respond again in the same unity".
On that last point at least, the two leaders saw eye to eye, with Kurz saying he wanted to beef up the EU's border force Frontex and give it more powers to "reduce the number of arrivals".