WASHINGTON—The Supreme Court agreed Tuesday to consider whether U.S. Border Patrol agents can be held liable for damages for shooting people on the Mexico side of the border.

The justices said they would hear a case on the 2010 cross-border shooting death of a 15-year-old Mexican boy who was playing in a cement culvert that separates El Paso, Texas, and Ciudad Juárez, Mexico. The teenager and his friends allegedly were playing a game where they would run up the concrete embankment and touch a fence in U.S. territory.

U.S....