Merkel has been asking her coalition to delay addressing the question until European Union leaders convene in Brussels to discuss immigration and other issues affecting the 28 members of the bloc on June 28 and 29.
Interior Minister Horst Seehofer is calling for Germany to turn back at its border migrants who have registered as asylum-seekers in other European countries.
She has meetings planned this week with the leaders of Italy and France, and an European Union summit is scheduled June 28-29.
Continuing on his anti-migrant tirade, Trump also said that crime in Germany was "way up" and called EU's immigration policy a "big mistake".
"We want to support Italy's desire for solidarity, and also hope that Germany receives understanding when it comes to European solidarity on the question of migration", she said in Berlin.
The sharp escalation of tensions between Merkel and her long-time Bavarian allies came with European Union nations once again at loggerheads over immigration, triggered by Italy's refusal this month to allow a rescue ship carrying 630 migrants to dock.
Merkel also distanced herself from plans of turning away migrants at the border, saying that such a move "could lead to negative domino effects" in Europe.
Even though both Merkel and Seehofer spoke of reaching some sort of a mutually-acceptable agreement and gave assurances of mutual support for "better management of migration and reduction of the migrants' inflow", their views on the future of the German migration policy still appeared to be conspicuously different.
The CSU fears that without a tougher migration policy, the far-right, anti-immigrant Alternative for Germany (AfD) party "will become stronger in Bavaria at the expense of the CSU", says Jürgen Falter, a political scientist and expert on Germany's far right.
The survey turns up the pressure on Merkel, who has faced a backlash for allowing into Germany more than one million people fleeing war and misery in Syria, Iraq and elsewhere since 2015.
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But just last month, Germany's Interior Minister Horst Seehofer released statistics showing that crime is at its lowest level in more than two and a half decades.
She said she would report back July 1 on the results of her negotiations, and that as far as she's concerned it's not yet clear what will happen if there's no European deal on the divisive topic.
Trump chimed into the debate with a Twitter taunt, charging that "the people of Germany are turning against their leadership as migration is rocking the already tenuous Berlin coalition".
"Whoever knows Europe, realizes this is no easy task", Merkel told reporters in Berlin yesterday after a meeting of her Christian Democratic Union's executive.
Germany's interior minister insisted that his party has no intention of bringing down Chancellor Angela Merkel and her government, amid uncertainty Sunday over chances of a solution to a bitter standoff over migration.
For her part, Merkel has expressed optimism about her leadership and her ability to manage Germany through the tensions.
Merkel emphasized the need for Germany's conservative parties to stick together, but she and Seehofer may only have delayed a head-on clash.
"After the European Council, the (CDU) party presidency will decide what to do next", she told a news conference.