City officials and the teachers union yesterday spent crucial negotiating time for a deal on a new teacher-evaluation system bickering over whether an independent mediator was needed.

The two sides have until midnight tomorrow to reach a deal and have it approved by the state Education Department or lose more than $250 million in funding.

Despite progress in the talks, Mayor Bloomberg called the United Federation of Teachers’ request for outside help “laughable.”

UFT President Michael Mulgrew said the city’s response “poses a serious threat to our ability to reach an agreement.”