Most of the city's roughly 1.1 million public school students take public transportation or walk to school.
Those who rely on the buses include 54,000 special education students and others who live far from schools or transportation. They also include students who attend specialized school programs outside of their neighborhoods.
Seeking a speedy end to the strike, a consortium of 20 bus companies filed two complaints with the National Labor Relations Board on Wednesday accusing the union of waging an unlawful secondary strike and of not bargaining in good faith.
"We are asking the NLRB for an immediate ruling," said Carolyn Daly, a spokeswoman for the bus companies.
James Paulsen, director of the NLRB's Brooklyn office, said the board is reviewing the complaints.
He said that if the NLRB finds that the union is pursuing an unlawful secondary strike, it will seek a federal injunction to halt the labor action.
The city doesn't directly hire the bus drivers and matrons, who work for private companies that have city contracts. The workers make an average of about $35,000 a year, with a driver starting at $14 an hour and potentially making as much as $29 an hour over time, according to Cordiello.
The city's last school bus strike, in 1979, lasted 14 weeks. Bloomberg said at his news conference, "I hope this is not going to last a long time but it's not going to last past June."
Associated Press writer Verena Dobnik contributed to this report.