WASHINGTON—The Supreme Court said Monday it would decide whether voters can bar state universities from using racial preferences in admissions decisions, a question that has divided federal appellate courts.

The appeal, involving a 2006 voter initiative from Michigan, is in some ways the mirror image of a separate case from Texas currently pending before the court. There, the justices are weighing whether the University of Texas' affirmative-action program, which the school says can give an edge to some minority applicants,...