State Education Commissioner John King was the first official to testify on the 2013-2014 budget this morning.

Albany — A day after Mayor Bloomberg declared the chances of a teacher evaluation deal with the city’s teachers union “impossible,” both sides confirmed this morning that they are returning to the table.

United Federation of Teachers President Michael Mulgrew first announced that talks were set to resume at the union’s legislative breakfast this morning, the Daily News reported.

The announcement comes hours before Mulgrew is set to testify before the state Assembly and Senate education committees about the 2013-2014 budget. He is among dozens of education officials and advocates who will make their case to the legislature about what they like and what they don’t like about Gov. Andrew Cuomo’s proposal.

Speaking now is John King, who asked for an extra $100 million in state aid, more money for early childhood education and $2.5 million for testing security and technology programs.

The renewed talks comes as Bloomberg faces mounting pressure from Albany to return to the table. New York City was one of six districts — out of 691 — that did not meet the Jan. 17 deadline, losing out on $240 million in state aid. It faces additional funding penalties if it does not show an effort to implement evaluations in the next two weeks.

Ratcheting up the pressure yesterday, Gov. Andrew Cuomo disputed Bloomberg’s testimony, which panned many of the state’s other district plans because many of them expired after one or more years. Cuomo told the Albany Times Union’s editorial board that Bloomberg’s critique was “factually incorrect” because the districts would still need to have teacher evaluation systems in order to qualify for state aid.

A source said Mulgrew first reached out yesterday afternoon, but it wasn’t to officials at the Department of Education, who he had been negotiating with directly when a deal fell apart two weeks ago. Instead, the source said, Mulgrew called City Hall, which he said has been pulling the strings in the negotiations.

“It was when they called the mayor that it blew up,” Mulgrew said yesterday, referring to the deal’s eleventh-hour collapse. “City Hall had told us that the DOE was fully authorized to make this deal so we negotiated with them. But it was the mayor.”

Bloomberg’s office didn’t return his call, but Chancellor Dennis Walcott did, the source said.

A city spokeswoman said Walcott has been reaching to the union since last week.

Both sides are now working out dates in which to hammer out the final details of a deal, which include when the deal would expire. They will likely seek to meet before Feb. 15, a deadline that the State Education Department has set for the city to show it is prepared to implement a teacher evaluation system. If the city misses that deadline, it will lose control of federal aid meant for low-income students.

Principals Union President Ernie Logan, in Albany to testify, said that he has not resumed talks with the city. The city is closer to a deal with the principals union than it is with the teachers.

“They haven’t called me yet,” Logan said.