The Alameda County Board of Supervisors has agreed to lease a former jail to the city of Oakland for a mere $1 a year in a plan to convert the facility into housing for the city’s ballooning homeless population, officials said Wednesday.

The controversial plan to turn Glenn Dyer Jail in downtown Oakland into homeless housing was floated last week by mayors of 14 cities in Alameda County, who said there were not enough shelter beds to handle the surging number of homeless people.

Oakland’s homeless population shot up 47% between 2017 and 2019 — one of the biggest increases of any California city — according to a recent count. And Alameda County’s homeless population shot up 43% during the same period.

The Board of Supervisors — which has a $937 million multiyear commitment to homeless services in the county — offered to provide the lease to the city on Tuesday and said it can begin hammering out the details of the plan with Oakland immediately.

“We need to all work together — cities and county — to find solutions,” Alameda County Board of Supervisors President Richard Valle said in a statement. “We have an urgent need to support those who are unsheltered and will work together with the cities of our county as they explore this ambitious and important effort.”

If the deal goes through, the Sixth Street jail that’s next to Oakland police headquarters and the county courthouse would be the first jail facility in the state to be converted into a homeless shelter.

The Alameda County Sheriff’s Office closed the jail in June to cut costs amid a declining inmate population.

The jail has plumbing, toilets and other facilities to accommodate more than 800 people.

Some homeless advocates, though, have pushed back on the idea, calling housing homeless people in a former jail “inhumane and immoral.”

Evan Sernoffsky is a San Francisco Chronicle staff writer. Email: esernoffsky@sfchronicle.com Twitter: @EvanSernoffsky