BATON ROUGE (AP) — A proposal to let Louisiana’s public colleges raise their tuition up to 10 percent a year narrowly won the backing of the state Senate on Friday.

As currently structured, the bill would let public colleges raise their tuition up to 10 percent this fall, instead of the 5 percent already authorized. In exchange, the schools would have to agree to work on performance improvements, like increasing admission standards, improving graduation rates and boosting efforts to get students into jobs.

The proposal — called the Louisiana GRAD Act — would set up voluntary six-year agreements with colleges. They’d have to start showing demonstrated progress on the performance benchmarks by the 2012-13 school year or lose the tuition increases.

Senators voted 26-13 for the measure, giving it the exact number of supporters it needed to reach the two-thirds bar required for passage.

The bill heads back to the House for approval of Senate changes. House Speaker Jim Tucker, R-Terrytown, sponsor of the proposal, said he was unsure if the House would accept the Senate changes or send the bill to a legislative compromise committee.

Supporters said the schools need to be able to raise tuition to help offset more than $250 million cuts in the last two years and to cope with another $300 million in reductions that are expected to be levied upon them in a year.

‘‘This bill is the opportunity to throw a lifeline to our colleges and universities in this state. I don’t think they’ve ever been in such a position as they are today,’’ said Sen. Ben Nevers, D-Bogalusa. ‘‘We will begin to dismantle the very economic engine of this state if we do not allow them some means to make up this lost revenue.’’

Opponents called the proposal a tax on students who can’t afford the price increase and said lawmakers should do more to restructure higher education to help cope with the revenue loss.

‘‘We can’t stand up and close some universities that need to be closed, merge some programs that need to be merged. We’re going to put it on the backs of mom and dad again, struggling to send their kids to college. Shame on us,’’ said Sen. Rob Marionneaux, D-Livonia.

Schools meeting the requirements could raise their tuition by up to 10 percent a year until they reach the average of similar schools in the South, or for LSU the average of state flagship schools around the country. After that, schools could increase tuition up to 5 percent a year or an amount equal to the growth in a national higher education price index, whichever is greater.