Speaking of the deal after terse negotiations on Monday, Mrs Merkel said: "We want on the one hand to set up transit centres in Germany and from there carry out returns in agreement with countries from which asylum seekers come and where they are already registered".
Although the move to appease the conservatives exposed her growing political weakness, Ms. Merkel will limp on as chancellor. "A majority of Germans back the chancellor".
Those who are rejected could be sent back to their European Union arrival country if that state agrees, says the joint CDU/CSU plan.
"The CSU's risky ego trip is paralyzing Germany and Europe", SPD chairwoman Andrea Nahles told journalists in Berlin. "There's no way out of the government for us and no way we'll end (the alliance) right now".
Crucially for Merkel, CDU lawmakers are backing her - so far. Bernd Riexinger, chairman of the Left Party, tweeted that "Transport centers are de facto mass internment camps".
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But he also struck a conciliatory tone, saying "there is an abundance of possibilities ... for compromises", without specifying what they were. "This so-called agreement leaves more questions open than it answers", she told Funke media group. It fears that tighter border controls by its northern neighbor could raise the number of migrants on its own soil.
Seehofer told party colleagues at an executive committee meeting on Sunday that discussions with Merkel had been fruitless, according to a party source.
Only once before has the cooperation been at risk - in 1976 when Bavarian leader Franz Josef Strauss thought the CSU would fare better in elections if it were free of the CDU.
He said he had spoken to Austrian Chancellor Sebastian Kurz by phone.
"So I think action in Germany to strengthen European interests is absolutely necessary", he said.
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Seehofer and Italian counterpart Matteo Salvini spoke by phone and agreed to meet for bilateral talks before a summit of interior ministers in Innsbruck on July 11.
Before the refugee crisis of 2015, Seehofer was mainly concerned with regional pet projects tolerated by Merkel, such as rewarding mothers who raise young children at home with state benefits and charging foreign motorists for using Germany's Autobahn. Seehofer is staying on as interior minister, a post that gives him federal border enforcement powers.
The row underlined the deep divisions that remain within Europe on how to deal with the migrants who have arrived in the last three years. However, there has been a surge in departures from Libya of migrants trying to cross by sea to Italy.
Seehofer had been in a showdown with the German Chancellor, offering his resignation on Sunday in protest over Merkel's reluctance to accept his plans to reverse the country's generous policies towards migrants.
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Knaus, director of the thinktank ESM, said the deal struck by the German government could have the unintended outcome of increasing migration into Germany, if for example the Greek government used the new agreement to unite families with people who had already arrived in Germany, while in return taking back some asylum seekers who have been finger-printed in Greece.