Days after London Breed took over as San Francisco’s acting mayor, her colleagues on the Board of Supervisors were already talking about appointing someone else to the position.

Breed, who is also the board’s president, was long seen as a contender to succeed Ed Lee in 2019. Lee’s unexpected death Tuesday vaulted her into the city’s top office, and if she can keep the high-profile job, she’ll have a big advantage in the June 5 election to finish the rest of his term.

But several supervisors and legislative aides are scrambling to block Breed. Two supervisors, Mark Farrell and Jane Kim, may challenge her for the seat in June, and four others — Aaron Peskin, Sandra Lee Fewer, Ahsha Safaí and Norman Yee — have endorsed Mark Leno for mayor.

“Obviously, this is on everybody’s minds,” said Peskin, a progressive who is close to Breed despite supporting Leno in the mayor’s race.

He and Supervisor Hillary Ronen are raising concerns about maintaining appropriate checks and balances in city government if Breed were to continue to serve simultaneously as acting mayor and board president. The dual role allows her to vote on legislation and then veto it.

The board has the choice of allowing her to remain acting mayor until the June election or appointing an interim mayor until then. If the supervisors were to choose from among themselves, that person — Breed included — would have to resign from the board.

If Breed were to continue as mayor in either capacity, she’d set the city’s policy agenda for six months, dominate news headlines and position herself not only to win in June but to clinch two additional terms and be at the top of city government for the next 10 years.

That worries City Hall insiders and politicians who are leery of repeating the history of 2011, when Lee, a former city administrator, was appointed to be a “caretaker” mayor, after some deft behind-the-scenes machinations by power brokers Willie Brown, the former mayor, and Rose Pak, who had ruled politics in Chinatown and died last year.

Lee was elected that November and re-elected in 2015, serving until his death Tuesday.

“There was no way Lee would have won that race in 2011 without having been an incumbent,” said former Supervisor Chris Daly, an outspoken progressive who was angered by Lee’s appointment.

It’s unclear whether the current board will rally behind a true “caretaker” mayor — someone qualified to keep the city running for six months who won’t seek the office in June. The deadline to file for the June election is Jan. 9.

Most names that have been floated so far are elder statesmen and women who are still part of the “city family”: former Mayors Brown and Art Agnos, former Supervisor and state Assemblyman Tom Ammiano and retired City Controller Ed Harrington, who was also the general manager of the Public Utilities Commission.

Others under consideration are high-level bureaucrats who have years of experience running departments, such as City Administrator Naomi Kelly and City Controller Ben Rosenfield. Whoever serves as mayor between January and June will have to handle labor negotiations with unions and oversee a complex budget-making process.

It’s unlikely that any supervisor would be able to amass six votes to get the interim seat, considering the 6-5 split between moderates and progressives on the board, and given that supervisors are barred from voting for themselves.

Brown, who is now a Chronicle columnist, scoffed at the idea that he could be appointed for his old job in City Hall.

“I’m not qualified,” he snapped. “Jurassic Park is where I now live, and they don’t let people like me out of Jurassic Park.”

Instead, he threw support behind Breed, whom he painted as a potential disciple to former Mayor Dianne Feinstein, now a U.S. senator.

“I’m betting she (Breed) will repeat what Feinstein did in 1978 when (George) Moscone was assassinated,” Brown said. “Feinstein went on to become a fabulous mayor ... ultimately emerging as a senior senator from California.”

Kelly resisted questions about a potential appointment Thursday.

“I frankly believe this is an inappropriate question at this time, as I am mourning the loss of my good friend Mayor Edwin Lee,” she said in an emailed statement. “I am focused on getting the city through this transition. ... I am focused on making sure the city continues to function appropriately and will continue to support the mayor and Board of Supervisors during this difficult time.”

Other power brokers, including veteran political strategist P.J. Johnston, touted Breed. Johnston called her “an incredibly capable person.”

“While I know she’s in a state of shock, it hasn’t diminished her capacity to lead,” Johnston said.

It appears the city’s moderate political establishment has coalesced behind Breed, who was already making public appearances as mayor Wednesday — her second day in office. That day she faced a throng of reporters at a news conference to promote a gun buyback program in the South of Market neighborhood.

On Thursday, she planned to join a holiday toy wrapping party at the African American Art & Culture Complex on Fulton Street. Her schedule of events this week seemed to build on the theme of helping out San Francisco’s disenfranchised communities — a fitting platform for a mayor who often stresses that she grew up in public housing in the Western Addition.

The longer she stays in the job, the more likely she’ll edge out potential moderate opponents, such as Farrell and Assemblyman David Chiu, and the greater threat she poses to Leno, a former state assemblyman and senator who entered the mayor’s race in May and has already raised $400,000.

But if some of Breed’s peers are intent on appointing someone else, others in City Hall say it would be a major political blunder to undercut San Francisco’s first female African American mayor.

“I think the women’s community will be watching,” said Deborah Mesloh, president of San Francisco’s Commission on the Status of Women, who has worked on campaigns for U.S. Sen. Kamala Harris, D-Calif.

“At this cultural moment in time, when there’s a revolution around equity for women, I can’t imagine an argument for London not to serve as mayor,” she said.

Rachel Swan is a San Francisco Chronicle staff writer. Email: rswan@sfchronicle.com Twitter: @rachelswan

Public services

Friday: Mayor Ed Lee will lie in state from 8 a.m. to 7 p.m. in the City Hall rotunda.

Sunday: A memorial celebration will be held at 3 p.m. in the City Hall rotunda.