Extending an olive branch to the City Council, Mayor Bloomberg has agreed to changes in the city’s budget process, the annual back-and-forth battle between the administration and the council that lawmakers have long criticized as a “dance.”

Responding to a request by Speaker Christine Quinn, the administration will disclose more detailed breakdowns of agency spending. The council had repeatedly derided department budgets for not including allocations by program, saying that as a result lawmakers could not properly evaluate whether a program was worth funding.

In an interview, Ms. Quinn called the accord “a very big win” that would allow the council to do a better job as a watchdog of city agencies.

The announcement yesterday comes two days before the administration releases its executive budget, and while the accord is not an overhaul of the often-contentious negotiation process, it could signal a smoother round of talks with the council this year. Lawmakers have often cited a lack of transparency in the budget process as a reason the council annually seeks to restore millions of dollars to programs the administration had proposed slashing.

In a statement, Mr. Bloomberg said increasing transparency was “the epitome” of good government.

Mr. Bloomberg’s consent to alter the process bolsters his strong relations with Ms. Quinn as she enters her fifth month as speaker. In contrast to her predecessor, Gifford Miller, who failed in his bid to unseat Mr. Bloomberg last year, Ms. Quinn has largely refrained from criticizing the mayor and has made numerous joint appearances with him.

“We’ve had a lot of success working together over the last four months,” Ms. Quinn said. Under the agreement, administration and council officials will jointly break down the budgets of two city agencies – the Administration for Children’s Services and the Department of

Small Business Services. The budgets will then be altered to provide more specifics for fiscal year 2008. Breakdowns of other city agencies will follow.

“We couldn’t change every agency tomorrow. It would be impossible,” Ms. Quinn said, citing the size of the annual budget. She said officials deliberately chose one social service agency and one economic. ACS, she said, was a logical starting point because it has already come under intense scrutiny after the January beating death of Nixzmary Brown, a 7-year-old under its care.

The executive director of Citizens Union, Richard Dadey, lauded the changes, saying they “would allow all government to better evaluate the appropriateness and the worthiness of many programs in the budget.”