Yet what she and other European leaders have quietly done over recent weeks is tighten asylum policy, restrict family reunions for refugees and mount campaigns to keep people from setting out for Europe. Balkan nations on the migrant trail that leads north from Turkey and Greece to Germany and Sweden have been encouraged to bar all but Syrian, Iraqi and Afghan refugees.

The new policy of steering or reducing the human flow culminates in a summit meeting of the 28-nation European Union with Turkey on Sunday. A measure of Ms. Merkel’s need to ease the refugee burden is that European leaders called the meeting in Brussels despite the terror alerts in that city after the deadly attacks in Paris on Nov. 13.

In exchange for better patrolling its Aegean coastline with Greece and cracking down on human smuggling, Turkey seeks three billion euros, about $3.2 billion, to help care for the 2.2 million mostly Syrian refugees it now houses. In addition, Europe is likely to pursue stalled negotiations on Turkish membership in the European Union and extend visa-free travel to many Turks.

Ms. Merkel used to oppose European Union membership for Turkey, and Europe has had many misgivings about human rights under Turkey’s president, Recep Tayyip Erdogan.