“Instead of just being need plus speed, it would move toward need plus college-ready,” said Branch, chairman of the House Higher Education Committee. “It would allow those who showed some college readiness to move to the front of the line.”

The measure was approved, 124-24, and faces a final House vote Thursday. A similar measure has already passed the Senate, and differences between the two will have to be negotiated if the Senate doesn’t OK the House’s changes.

The prioritization matters because the program is not fully funded — about 50 to 75 percent of eligible students who apply get a grant. And the House’s proposed budget would cut funding even more, so that no new eligible incoming freshmen would be able to receive a grant, affecting about 60,000 students. Senators hope to cut the program less drastically.

An amendment added to Branch’s bill by Rep. Joaquin Castro, D-San Antonio, would provide a way for students who cannot receive a grant because of a funding shortage during their first year of school to be eligible for funds later.

Some critics say that the measure may hurt minority enrollment in state universities. More than 60 percent of grants currently go to minorities.

And Democratic representatives argued that there is no need for the change.

“This TEXAS Grants program has worked well,” said Rep. Helen Giddings, D-Dallas. “College participation is up and they are earning degrees.”