NEW YORK (CBSNewYork) — Union leaders claim the city has shot down their latest offer to end the two-week-old strike by school bus drivers and matrons.

Michael Cordiello of Local 1181 of the Amalgamated Transit Union said the union had requested a “cooling off” period of up to three months — during which drivers would have gone back to work — if the city was willing suspend the opening of the special education bid and come to the table.

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The city rejected the proposal, Cordiello said.

“Judge Milton Mollen brought this to the city’s people, to the mayor’s attorneys and the idea was rejected,” Cordiello told reporters including WCBS 880’s Marla Diamond.

The walkout began on Jan. 16, triggered by the city’s plan to put bus contracts out to bid to lower costs.

Cordiello has said the drivers will strike until Mayor Michael Bloomberg and the city agree to put a job security clause back into their contract, but the Bloomberg administration has insisted that it cannot and adding such a clause would be illegal.

Earlier this week, the mayor organized a meeting between school bus companies and the union in an effort to resolve the strike, however, no city representatives were present.

“He needs to come to the table and resolve this in an intelligent and useful manner,” said Cordiello.

Bloomberg has stressed from the start that this is an issue between the employers and their employees. The city contracts with private bus companies and says it must seek competitive bids to save money.

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“The union is irresponsibly holding our students and city hostage over issues that can only be resolved by negotiating directly with the bus companies,” a city spokesperson said in a statement.

Bus contracts costs the city $1.1 billion each year, according to the mayor’s office. The city said postponing contracts that were released for bid in December would ensure that existing contracts remain in place next year.

“I am extremely disheartened that Mayor Bloomberg continues to abdicate his responsibilities to the city of New York,” Cordiello stated. “The mayor has just as much of a responsibility to end this strike and get our city’s safest and most experienced drivers and matrons back on the road as anyone else. Instead, Mayor Bloomberg continues to put the safety of the city’s children at risk by ignoring decades of history and the actions of his predecessors by falsely claiming he is unable to come to the table.”

“We need all parties to come to the table and negotiate in good faith,” said Vincent Alvarez, President of the New York City Central Labor Council. “The city could end this tomorrow, but instead continues to take us down this irresponsible path. By rejecting the union’s offer to return to the job while working toward a long-term resolution, the city is demonstrating a total disregard for children, parents, workers and all New Yorkers.”

“There are real consequences as a result of the Mayor’s refusal to come to the table and negotiate,” said Mario Cilento, President of the New York State AFL-CIO. “Children and parents are suffering, and working class families have been forced to fight for their livelihoods. It’s time for the Mayor to stop being unreasonable, and start working with the union to get the workers back on their routes providing safe and reliable service to New York City school children.”

The strike has idled more than half the city’s school buses, forcing thousands of students to find other ways to get to school.

Some bus companies decided to hire replacement drivers on Tuesday to help get special needs students to class.

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For more information about the strike for parents and students, click here.