Interior Minister Horst Seehofer has proposed turning back refugees who already were registered in other European countries, as part of a lengthy "master plan" on curbing unauthorized immigration.
Seehofer, who heads the Bavaria-only sister party to Merkel's center-right Christian Democratic Union, is under pressure to take a hard line ahead of Bavarian state elections this fall.
Seehofer told reporters Wednesday that he wants an agreement this week, and backed Austrian Chancellor Sebastian Kurz's idea for an Italian-Austrian-German "axis" to tackle illegal migration.
Both Austria and Italy have seen parties with strong anti-migrant policies come to power since 2015, when Europe experienced a sharp rise in the number of people seeking shelter from conflicts and economic hardship elsewhere in the world.
The chaotic influx and some countries' harsh treatment of the migrants prompted Merkel to open Germany's borders, a decision she has strongly defended as a necessary response to a humanitarian crisis.
Merkel insisted Wednesday that she wants a "solution for all of Europe" rather than one that works only for some countries.
"This may sound like squaring the circle, but Europe needs to stick together," she said after a meeting with Belgian Prime Minister Charles Michel. "It's very, very important to me that Germany doesn't act unilaterally, but that everything is done in an orderly way, that it's coordinated and underpinned by an agreement."
The German leader added: "For me, the way we treat the migration problem is something of a litmus test for the future and cohesion of Europe, and the needs of every country need to be taken into account."
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