BOTOVO, Croatia—The European Union could get an agreement with Turkey to try to stem the flow of migrants in the coming weeks, Croatian Foreign Minister Vesna Pusic said Monday, as her country struggled to handle the rising numbers heading for the more affluent West.

EU leaders are expected at a meeting in Brussels this week to look at ways to encourage Turkey—financially and otherwise—to crack down on people smugglers who have a lucrative business of shipping migrants from Turkey on a route through Europe into Germany, officials...