City voters will have a chance to extend for a decade — or halt altogether — free classes at City College of San Francisco, now that the Board of Supervisors has agreed to place the question on the Nov. 5 ballot.

The supervisors voted 7-1 Tuesday to put a charter amendment before voters that would pay for 10 years of no-cost classes for all city residents, starting in 2020. Supervisor Catherine Stefani cast the lone “no” vote, and Supervisors Malia Cohen, Aaron Peskin and Katy Tang were absent.

The free tuition program began last year and is scheduled to expire in June. College and city officials expect to reach an agreement to fund the program until voters decide what to do, said Supervisor Jane Kim, who has championed the Free City program and the 2016 ballot measure that pays for most of it.

“We have one of the fastest-growing economies that is the envy of cities and states around the nation, and the fastest-growing income gap,” Kim said. “The great equalizer has always been education.”

It was the final board meeting for Kim, who is leaving office because of term limits.

The proposed charter amendment would establish the Free City College Fund, which would be administered by the city’s Department of Children, Youth, and Their Families. The funds would come from a set-aside — money earmarked for a specific purpose — in the city’s general fund.

If voters approve, the city would appropriate $15 million to $16.4 million a year for free classes at City College for three years, starting in fiscal 2020-21 After 2023, the city controller would have the authority to add or subtract funding each year, depending on how much discretionary revenue the city has. The charter amendment would need a simple majority vote for approval.

Free City College is funded thanks to Proposition W, the 2016 ballot measure that raised a city transfer tax on residential and commercial properties that sell for at least $5 million. It raised about $28 million this year and is not set to expire.

The money from Prop. W goes into the city’s general fund, but supporters of the tax were clear at the time that it would be spent to fund residents’ tuition at City College. The tax has come up $5.4 million a year short of full funding, however, forcing City College to cover the rest.

The free-tuition program means San Francisco residents don’t have to pay $46 per unit to attend classes. The program provides grants of up to $500 a semester for textbooks and other costs of attending college for low-income students who already qualified for free tuition.

Stefani, the only supervisor to vote against putting the charter amendment on the ballot, said she believes in a free City College but not in funding it through a budget set-aside.

“I support free City College for all San Franciscans, but want to see it done in a way that maximizes federal and state aid for students and creates an economically sustainable City College,” she said. “Set-asides are a fiscally irresponsible practice that limits our ability to effectively run the government.”

The measure would create the city’s 20th budget set-aside, according to a report by the city controller.

Stefani also said City College has failed to require students to exhaust all other sources of funding, such as Cal Grants and federal Pell grants, before tapping into the Free City money.

But Supervisor Rafael Mandelman, a former president of the City College Board of Trustees, said it is important to get students into school first. Then the college can work with them to tap other aid sources, he said.

“This charter amendment is important because it settles the question of whether this is a long-term program that its students can rely on,” he said.

Under the proposed charter amendment, the city could freeze its contribution in any year when it runs a budget deficit of at least $200 million. The Board of Supervisors would also have the authority to adjust contributions to the Free City fund by two-thirds vote if, for example, it decided City College could pay for the program itself.

Separately from Tuesday’s vote, City College Chancellor Mark Rocha told Mayor London Breed this month that he would like to reach a deal with the city to make changes to the free-tuition program, including steering more money toward low-income students.

“After nearly two years of administering the Free City program in its current form, we are keenly aware that (it) isn’t adequately serving our students, particularly our most vulnerable students,” Rocha said in a letter to Breed.

As for the proposed charter amendment, a spokesman for Breed said, “The mayor supports the free City College program. and she will be looking at the measure to determine whether it aligns with her priorities.”

Trisha Thadani and Nanette Asimov are San Francisco Chronicle staff writers. Emails: tthadani@sfchronicle.com, nasimov@sfchronicle.com Twitter: @TrishaThadani @NanetteAsimov