New Jersey Gov. Phil Murphy is gearing up to make the state’s community colleges tuition-free with $45 million in grants for low-income students that will start in the second semester of the coming school year.

The grants would fully cover tuition costs for an estimated 15,000 full- and part-time students who are already receiving some financial aid and whose families earn an average annual income below $45,000, the governor’s office said.

Mr....