Updated 07/18/11 – 4:34 p.m.

CHICAGO (CBS) — The Emanuel administration and labor union leaders were back at the bargaining table on Monday, trying to work out a deal to avoid 625 threatened layoffs of city workers.

Mayor Rahm Emanuel has long said he wanted a partnership with city employees and their unions, but then he seemed to pull the rug out from under them on Friday by announcing plans for 625 layoffs when the unions missed his deadline for cost saving ideas.

But the two sides sat down again on Monday to try and hammer out a deal and CBS 2 Chief Correspondent Jay Levine has been told a member of the mayor’s team made a surprising admission.

The stage was set for tough talk from both sides, with labor leaders furious at Emanuel for letting them have it with both barrels on Friday.

But instead of following their boss’s aggressive lead, the Mayor’s team seemed to back off a bit in talks about how to save money and jobs at the same time.

At the same time the Mayor was announcing that private companies would be taking over recycling duties from city workers, at a savings of millions of dollars a year, labor leaders were gathering to meet with the mayor’s team about cost-saving work rule changes to avoid more job losses.

As WBBM Newsradio 780’s Regine Schlesinger reports, the mayor says labor faces a clear choice – accept work-rule changes to rein in city costs, or sacrifice jobs.

LISTEN: Newsradio 780’s Regine Schlesinger reports

He says his obligation is to the taxpayers, not the payrollers.

“For too long, what’s happened in Chicago has been the process of an inside game, benefiting the few and those with influence,” Emanuel said.

“I still believe the choice should be work rule reforms, not workers’ jobs. That’s not a choice I wanna make,” Emanuel added. “My team is gonna lay out – again, specifically – nine ideas on reforms.”

Emanuel said last week that he will lay off 625 workers as part of a plan to close a $10 million budget gap.

He said he will outsource airport and library janitors, benefits management workers and workers at the water billing call center. He also plans to cut seasonal transportation department workers. Under contract, the workers are given a 30 day notice of a planned layoff.

In making that announcement Friday, Emanuel said the city’s labor unions had already had time to decide on contract changes that could reduce personnel costs, such as ending time-and-a-half pay for duties before a shift.

While the mayor had set the tone for Monday’s meeting with the unions in his layoff announcement on Friday, the meeting with labor leaders apparently took a surprising turn.

“There have been no notices of layoffs, in fact there was a much different tone today coming from the administration,” AFSCME Council 31 Executive Director Henry Bayer said after the meeting.

In fact, one source said that the Mayor’s representative, Mark Angelson – who refused to answer any questions when he left the meeting – actually admitted the “Mayor mishandled, misplayed (Friday’s announcement)”, and that “Rahm now gets it.”

Sources close to the Mayor disputed the interpretation that Emanuel had a change of heart, admitting only that there might have been misunderstanding about the deadline for labor’s response to the Mayor’s proposed reforms – which they apparently still didn’t have ready on Monday

“We didn’t discuss any … specific efficiency today,” Bayer said. “We indicated to the city that we will be contacting them to let them know.”

Bayer said AFSCME has been working with the Chicago Federation of Labor, which has commissioned a study to come up with a set of efficiencies to help the city save money and avoid layoffs.

Make no mistake about it, Bayer is one tough hombre, but he might have met his match in the mayor. This might be the case of one side testing the other face-to-face, eye-to-eye, waiting to see who blinks first.

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