WASHINGTON—President Barack Obama has ordered a delay in an administration review of deportation policy in hopes that the House of Representatives will take up the larger immigration overhaul this summer, in what is seen as the last chance for action this year and maybe beyond.

The decision puts the focus squarely on House Republicans, who must decide whether to bring legislation to the House floor before Congress breaks for its annual August recess…

The statement came from the National Hispanic Christian Leadership Conference, National Immigration Forum, Service Employees International Union, Sojourners, the Leadership Conference on Civil and Human Rights, and the United States Conference of Catholic Bishops Committee on Migration.