ATHENS—The Balkan trail taken by migrants was completely shut down Wednesday, officials from several countries said, adding to pressure on the European Union to complete a deal with Turkey to stem the flow of people.

In recent weeks, Austria and Balkan countries to its south have coordinated a tightening of their borders, moves that have effectively limited the number of people able to continue north from Greece to fewer than a hundred a day. The restrictions have left tens of thousands of migrants stranded in Greece after...