BERLIN — Hours after Chancellor Angela Merkel’s conservative bloc agreed to measures that would tighten Germany’s southern border, Austrian leaders announced plans on Tuesday for similar actions that would further threaten Europe’s system of free movement.

In a last-ditch effort to save her government after weeks of bitter infighting with rebellious coalition partners, Ms. Merkel agreed late Monday to a plan to set up camps along the 500-mile border it shares with Austria to screen newly arriving migrants. Anyone found to have asylum applications pending in another European Union country would be sent back there.

The deal still needs the approval of the center-left Social Democrats, the third party in Germany’s governing coalition along with Ms. Merkel’s Christian Democrats and their Bavarian partners, the Christian Social Democrats. The Social Democrats have opposed such plans in the past, criticizing the arrangements as “mass internment camps,” a phrase with obvious historical overtones for Germans.

The plan further stipulates that if a European partner refused to take back asylum applicants, they would be sent back “on the basis of a deal with Austria,” though it is not yet clear what that deal would entail.