The budget proposal threatens to strip state funding from private institutions that fail to keep their yearly tuition increases under $500 or the higher education rate of inflation, whichever is greater. (That rate is about 2.3 percent.) Starting in 2018, New Yorkers who choose to attend private colleges that do not comply with the cap would not receive tuition grants from the state’s tuition assistance program, known as TAP, which covers students with incomes up to $80,000 a year.

Mr. Cuomo did not mention the proposal in last week’s budget briefing. About 90,000 students at private colleges currently qualify for state tuition assistance, according to the governor’s office.

“You’re not giving the student the money that would help them with the choice of where they want to go to school,” Mary Beth Labate, the president of the Commission on Independent Colleges and Universities, which represents many of the state’s private, nonprofit institutions, said in an interview.

The change will likely hit hardest the small private colleges that recruit students mainly from New York State and rely mostly on tuition for their operating budgets. With State University of New York and City University of New York campuses becoming free for many more students, private liberal arts enrollment is probably likely headed for a drop.

Private, nonprofit colleges “leverage their aid dollars to make education possible for needy students,” said Margaret L. Drugovich, the president of Hartwick College, a small liberal arts college in Oneonta. “The governor’s proposal limits the choices for these students. Why should we do this? What societal goal does this advance?”