Mr. Steinmeier stressed that Germany now expects 800,000 migrants to arrive this year, and added, “None of these will be taken in by others, so I ask for understanding, that we discuss not only those who are already here, but a fair quota of distribution of those who are still on the way.”

His statement during talks in Prague with Hungarian, Czech, Slovak and Polish counterparts suggested that neither he nor the foreign minister of Luxembourg, which holds the rotating presidency of the Council of the European Union, had persuaded the former Communist countries, which have not been immigrant destinations, to adopt quotas to house 160,000 migrants as suggested this week by the European Commission president, Jean-Claude Juncker.

The Prague talks were held in anticipation of a meeting on Monday of European interior ministers to discuss the Juncker proposals, and there were indications on Friday that heads of state and government might have to hold a summit meeting if no progress is made then. Officials in Berlin, speaking on condition of anonymity as they groped for a solution, even mentioned using majority voting — and not the traditional consensus — to decide the matter.

That would be highly unusual in an affair so central to Europe’s destiny.

But Mr. Steinmeier indicated that Berlin is keenly aware that failure to control the situation might tip the popular mood in Germany, and other countries like Sweden and Austria, where the welcome for the migrants has so far been mostly warm.