NEW YORK—U.K. Prime Minister David Cameron said Europe’s migrant crisis was complicating his efforts to change Britain’s relationship with the European Union, a central policy goal of his second term in office.

In an interview with The Wall Street Journal on Monday, Mr. Cameron said the crisis made it more challenging for him to make the case for staying in the European Union to the British people, who were watching waves of refugees and other migrants arrive in Europe from Syria and elsewhere with concern.

“It...