President Emmanuel Macron and Chancellor Angela Merkel met on June 19 in Meseberg, near Berlin.
German Interior Minister Horst Seehofer, who also leads the CSU, may be tempted to use his powers to implement his party's call for more powers to turn away refugees registered in other European Union states.
At the centre of the showdown between Merkel and Seehofer, who are bitter rivals after the chancellor's decision to allow nearly a million refugees into Germany three summers ago, is disagreement over whether to allow those who have already registered in another European Union country entry to Germany.
The Merkel party, Christian Democratic Union, CDU, and Seehofer, Christian Social Union, CSU, are twinned formations for decades and toger y form conservative bloc that shares government in Berlin with Social Democrats.
The CSU's top priority is a hard October state election in Bavaria in which it is trying to tamp down support for the anti-migration Alternative for Germany party.
Three years after her decision to open Germany's borders to migrants fleeing war in Syria and Iraq and misery elsewhere, Merkel is still struggling to find a sustainable solution to end the grumbling from her Bavarian allies CSU over her liberal refugee policy.
European stocks, and particularly the German market, have been hit once again over recent days by the possibility of political turmoil in Germany caused by disagreements within the ruling coalition over the country's migrant policy.
There is, she insisted, "nothing automatic" about what happens next.
"We want more sovereignty and unity", Macron said.
"Germany also has a head of government", he said of Merkel, saying it was she who was "responsible before her people and her parliament".
Stressing that Germany must change direction on immigration, Seehofer said: "I can not say that we have a grip on the issue". "But we stand by our position that, if this does not succeed, turning people back immediately at the border must be possible".
"We're hoping to obtain a maximum of agreements on Tuesday, and in late June", a source in the French presidency told AFP.
Seehofer said his party was keen to find a way to limit the number of asylum seekers arriving in Germany.
The spat over immigration has laid bare the deep tensions in a fractious German government that took office only in March, after almost six months of postelection haggling, and exposed the limits of Merkel's authority.
But Merkel is firmly opposed, warning that it would leave countries at the EU's geographic southern periphery alone to deal with the migrant influx.
"Angela Merkel's coalition is coming under increasing strain over the migrant issue".
Some 49% of voters surveyed last week by pollsters Civey for Die Welt newspaper said they either "definitely" or "sooner" place trust in Seehofer to solve the perceived asylum crisis compared with 32% who said Merkel has the right approach.
The rules require migrants to apply for asylum in the first country they enter-a policy France has used to defend its own efforts to keep people from trying to cross over from Italy. More than 1 million migrants came to Germany in 2015 and 2016, though numbers have since dropped sharply.
The figures showed that police reported 5.76 million crimes in 2017, down five percent from the previous year.
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