The Obama administration plans to restore federal funding for prison inmates to take college courses, a potentially controversial move that comes amid a broader push to overhaul the criminal justice system.

The plan, set to be unveiled Friday by the secretary of education and the attorney general, would allow potentially thousands of inmates in the U.S. to gain access to Pell grants, the main form of federal aid for low-income college students. The grants cover up to $5,775 a year in tuition, fees, books and other education-related...