A strike was to begin as early as Wednesday, but Teamsters Local 445 agreed to extend its deadline at the request of federal mediator Kathy Murray-Cannon.
The key issue is pay, according to a union spokesman.
The county urged passengers to seek alternative transportation in case of service disruptions during a strike.
The union has agreed to notify the public at 5 p.m. the day before a strike begins.
The bus drivers' starting pay is $18.83 per hour, and tops off at $21.97 per hour, according to the union.
Union members learned of the delay Tuesday in person and through newsletters.
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