According to the L.A. Times, some would receive an incentive of at least $15,000 to leave.
The city's executive employee relations committee, which is comprised of Villaraigosa and four city council members, has been negotiating a potential deal with a coalition of L.A. city unions.
The newspaper reports that city leaders are hoping to reduce the workforce by 2,400 employees.
The plan would also cut pay raises for most workers for the next two years.
The proposal is a shift from the council's decision earlier this month to force thousands of workers to take every other Friday off without pay, but it will likely not save the dozens of city employees set to be laid off by the first or second week of July.
They city says 76 positions are slated to be eliminated in the next couple of weeks.
The mayor's committee is set to meet again on Tuesday.