But the crisis shaking the new administration has come from within Merkel's own conservative family, as her interior minister has pushed for Germany to turn back unilaterally some migrants at its border.
Seehofer last week defied Merkel to announce a tough new immigration policy that would see some asylum seekers turned away at the borders of Germany. "But we stand by our position that, if this does not succeed, turning people back immediately at the border must be possible".
However it ends, the spat has laid bare the limits of Merkel's authority in a fractious government that took office in March after almost six months of postelection haggling.
Seehofer said he had told fellow leaders that "we're not out of the woods yet".
Merkel's junior coalition partner - the Bavarian Christian Social Union (CSU) - is under pressure from the nationalist Alternative for Germany (AfD) party, which became Germany's largest opposition party in the elections a few months ago.
Chancellor Angela Merkel said she and French President Emmanuel Macron agreed on Tuesday to create a euro zone budget charged with boosting investment in the currency bloc and promoting economic convergence between its 19 member states. The CSU lost thousands of voters to the far-right Alternative for Germany (AfD) party in federal elections past year, and is fearful of further losses.
"The overwhelming majority of the German population supports this idea and this concept, and that's why we want to provide support today for implementing it", he said.
It's the first time Trump has openly criticised Germany's immigration policy since he took office past year, although he was previously critical of their immigration policy during the presidential campaign when he called Angela Merkel's decision to leave the country's borders open to Syrian refugees as "insane".
Bavarian governor Markus Soeder and the party's top federal lawmaker, Alexander Dobrindt, have been even more vehement than Seehofer in demanding immediate action on migration.
The president on Monday also said that children are being used as a means for criminals to enter the United States, and he cited the crime in countries south of the United States' borders.
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".
Trump then trained his sights on Germany, which has experienced problems with assimilation since it took in millions fleeing the Middle East during the European migrant crisis.
Merkel's liberal approach to immigration has been controversial in Germany, however, particularly on the right.
"Germany also has a head of government", he said of Merkel, saying it was she who was "responsible before her people and her parliament".
She asked the CSU to give her two weeks to come up with deals similar to one agreed between Turkey and the European Union in 2016, which means migrants arriving in Greece are sent back to Turkey if they do not apply for asylum.