It is far from certain whether the bus drivers who work for the companies that serve the city will stage a strike, since not all of them are unionized. Michael Cordiello, the president of Local 1181, which represents 9,000 affected members, did not return phone calls on Friday seeking comment.

The issue dates back to 1979, when job protections were first put into transportation contracts after the last bus strike, which lasted 13 weeks. It arose anew in 2011, after the city issued a request for bids for the transportation of 14,000 preschool students with special needs. Mr. Cordiello then said a strike was probable. But days later, he said that contract negotiations would be reopened and no strike took place.

On Friday the city put out another set of bids, this time for the transportation of 22,500 children in kindergarten through 12th grade who have special needs and require special transportation. The bundle of contracts at stake cover 1,100 bus routes, or about one-sixth of the Department of Education’s total routes, and will expire June 30.

What has again piqued the union’s ire, according to the city, is that the new round of bids, like the old ones, do not require new companies picked for the job to hire, by order of seniority and at the same rate of pay, bus drivers and other workers from the companies that lost the contracts.

Mr. Bloomberg said that the Court of Appeals, the state’s highest court, in a ruling last year, legally barred the city from including employee protection polices in transportation contracts.