The Oakland City Council next month will reconsider a controversial proposal to give city funds to several private job-training organizations.

The proposed ordinance, authored by City Councilwoman Desley Brooks, would take a percentage of various public funds and give the money to several groups that provide vocational training to Oakland residents.

Last week, City Attorney Barbara Parker said using some of the funding sources the Brooks proposal identified — such as specific bond measures — “would violate state and federal laws, the terms of the voter approved bond measures, and the express terms of the bond documents.” Others, like parking revenue, could be fair game.

But Brooks on Tuesday deleted language that specified three potential sources of the funds — the gas tax and two voter-approved bond measures. She said doing so would take care of any legal issues.

“There is a crisis in our community,” Brooks said in a letter to her colleagues on the council. “Our residents are being pushed out of our city. They are being locked out of the construction boom that is taking place.”

She criticized Parker’s legal memo as “long on opinion and short on law.” City lawyers said there were still problems with the proposal, even with the new tweaks, prompting Brooks to accuse them of “gamesmanship.”

Under the revised proposal, the organizations would receive 5 percent of capital improvement project costs, 5 percent of parking revenue and 5 percent of development services — a fund into which developers pay for city services such as building plan checks. City contractors would also be charged 30 cents per hour worked and would be subject to late fees if they do not send checks on time.

The groups identified in the proposal to receive the funds are the Cypress Mandela Training Program, which has operated at a deficit in recent years; Men of Valor, which is run by leaders of religious organizations; and the Laborers Community Training Foundation, which is in San Ramon and connected to a politically active union.

Other centers that “provide accessible, high-quality training and employment services to local residents and employers” could also qualify for the funds.

Ed Gerber, chairman of the citizen Budget Advisory Committee, said even with the deleted language, dedicating funds to flow directly to independent organizations without any oversight remained problematic. If the city wants to fund nonprofits, he said, it should use a more standard competitive bidding process in which multiple groups could apply and guarantee “deliverables.”

The council voted 4-3 to bring the item back June 5 with additional analysis, including how much money exists to give to the groups. Council members Brooks, Rebecca Kaplan, Noel Gallo and Lynette Gibson McElhaney voted yes; Dan Kalb, Abel Guillen and Annie Campbell Washington were opposed; Larry Reid was absent.

Kimberly Veklerov is a San Francisco Chronicle staff writer. Email: kveklerov@sfchronicle.com Twitter: @kveklerov