The disagreement between Angela Merkel's Christian Democrats and their Bavarian Christian Social Union sister party threatens the future of her coalition.
Veteran CSU politician Hans-Peter Friedrich fueled the discussions, stating in an interview on German television that his party "stood united behind Interior Minister Horst Seehofer" in the row with its allies, but there were "no demands to end the cooperation with Angela Merkel". Merkel has warned that such a move could shift the burden onto countries such as Italy and Greece. "Germany can not wait endlessly for Europe, but must act independently", Markus Soeder, the CSU's Bavarian "prime minister", said on Thursday.
Speaking alongside German Interior Minister Horst Seehofer, a Bavarian conservative who shares the Austrian chancellor's views on tightening up Europe's borders, Kurz said fighting illegal immigration would be a top priority for Vienna's presidency of the Council of the European Union in the second half of 2018.
It cost her dearly in the 2017 election as her party bled voters to the far-right Alternative for Germany.
The Augsburger Allgemeine newspaper quoted an unnamed senior CSU lawmaker as saying the Bavarian party was considering leaving the parliamentary bloc.
He says German border police should turn back all asylum seekers who lack ID and those already registered in another European Union country.
Seehofer, the former premier of conservative Bavaria state, has always been one of the fiercest critics of Merkel's decision to open Germany's borders at the height of Europe's migration crisis in 2015.
Merkel believes that Berlin should not take unilateral asylum measures and argues that a measure affecting free movement in EU must be adopted in Brussels, particularly at forthcoming European Council on 28 June.
The CSU's Bavarian state premier Markus Soeder charged that the ongoing "asylum tourism" meant that, as a government, "we're nearing the endgame when it comes to our credibility".
For Seehofer, adopting a strong stance on migration is politically crucial as he and the CSU gear up for state elections in Bavaria set for October 14.
Merkel has proposed that asylum seekers already rejected by Germany could be turned back at the border.
If Seehofer goes ahead on Monday, that would leave Merkel with two choices, both of which could spell the end of her 13-year rule.
In that scenario, Merkel could lead a minority government with the third party in the coalition, the Social Democrats (SPD), or call fresh elections that would likely benefit only the AfD, he said.
SPD Finance Minister Olaf Scholz added: "The task of governing our country is not an episode of Game of Thrones, but a very serious matter".
The same day he met Austrian Chancellor Sebastian Kurz, who announced at a joint press conference that the interior ministers in Vienna, Rome and Berlin had formed an "axis of the willing" to combat illegal immigration.