The crisis that has raised questions over the future of Merkel's 3½-month-old government pits Interior Minister Horst Seehofer and his Bavaria-only Christian Social Union against Merkel, head of its longtime sister party, the Christian Democratic Union.
The Austrian Chancellor, Sebastian Kurz, has said that Germany's new migration agreement, which ended a political crisis in Angela Merkel's government, could force Austria to introduce tougher border checks on Italy and Slovenia.
The deal still requires the consent of Merkel's other coalition partner, the centre-left Social Democrats, to become government policy.
Germany's constitution prevents federal ministers from outright quitting their positions, and must request to be dismissed by the German President.
German Minister of the Interior Horst Seehofer insisted on his plan to turn away asylum seekers at the border with Austria registered in other European countries, as he rejected EU deals reached last week by Merkel as inadequate.
In high-stakes crisis talks on Monday, Ms. Merkel put to rest for now a unsafe row with a longtime rival, Interior Minister Horst Seehofer, that had threatened the survival of her shaky 100-day-old coalition.
The agreement sets up zones along the border with Austria to facilitate quick deportations for migrants not allowed to seek asylum in Germany.
Merkel called the proposal a "good compromise".More news: House soundly rejects last-ditch effort to pass immigration reform bill
Seehofer is threatening to resign amid a growing divide with Chancellor Merkel over migration policy. He insisted the CSU doesn't want to break up the conservative partnership.
The BBC's Jenny Hill in Berlin says Mr Seehofer may have overplayed his hand by issuing his ultimatum, only for Mrs Merkel to return from Brussels on Friday with an EU-wide strategy and bilateral agreements with more than 10 countries.
The leadership of Merkel's party approved a resolution Sunday stating that "turning people back unilaterally would be the wrong signal to our European partners".
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. "A majority of Germans back the chancellor".
The strongest voice in opposition to mass migration in the Bundestag is the Alternative for Germany (AfD) party, who have 94 seats to the governing coalition's 246.
Political stability was upset by Merkel's 2015 decision to keep borders open to migrants and refugees arriving from the Middle East via the Balkans, Hungary and Austria.
Nevertheless, the anti-refugee, anti-Islam Alternative for Germany (AfD) was propelled into federal parliament for the first time past year by outrage over immigration, leading to months of paralysis while Merkel struggled to put together a workable coalition.