WASHINGTON—The Supreme Court will consider whether the Constitution allows a Catholic organization to reject same-sex couples as foster parents in a city-funded child-welfare program, despite a local ordinance prohibiting discrimination based on sexual orientation.

The case—a potential watershed for religious rights—pits the city of Philadelphia against Catholic Social Services, an arm of the Archdiocese of Philadelphia that has been placing foster children in homes since 1919. The city suspended referrals to the organization...