When Mayor Michael Bloomberg presented his preliminary budget to the city four months ago, he extended an invitation: “If you come up with a cut you don’t like, just tell me where else we can cut, and tell me how the person whose ox is going to be gored is going to handle that.”

Until recently, City Council members and advocates who oppose many of the mayor’s cuts, particularly those in areas like education, housing and children’s services, have called for raising taxes as the solution. But with Bloomberg and Albany policymakers firmly opposed to that tactic, at least a few groups have pinpointed pots of money at city agencies that they say the mayor has overlooked and can help close the $5 billion budget gap while preserving critical services.

Most remarkably, a homeless services organization has asked the city to cut back one of its own contracts in the hopes of freeing up cash for shelter services like maintenance, child care and case work–services which will be greatly reduced if the mayor goes ahead with his plan to decrease shelter contracts by 5 percent. The savings can be found in shelter medical services, argues Lauri Cole, executive director of the ASPHA/Tier II Coalition, which represents 57 nonprofit providers of emergency housing and homeless services. On-site nurses are on duty during hours when they are rarely needed, she said. By trimming those hours by roughly one-third at three family shelters and at the city’s Emergency Assistance Unit–the intake center for homeless families–Cole and some of her members estimate the city could save about $1.2 million.

The city has yet to weigh in on this proposal, but the plan does have its opponents. “We are seeing a lot more families coming in with serious medical or psychiatric problems,” said Patrick Markee of the Coalition for the Homeless. “To reduce medical services at this time doesn’t seem to be wise.”

Still, Cole stands by the proposal as a means for funneling cash elsewhere: “None of our members would ever suggest any cost-saving measure that threatened to harm homeless families in any way.”

Other proposals, while perhaps a little more self-serving, have also been made. In a May report, the city’s municipal worker’s union, DC 37, identified roughly $300 million in cuts that weren’t in the mayor’s budget–like the canceling of contracts that pay outside agencies and consultants to do city work, an idea the union’s executive director, Lillian Roberts, says Mayor Bloomberg is taking seriously.

City Project, a budget watchdog, has found some potential savings as well. Pointing to an early 1990s pilot project in Brooklyn that saved more than $10.5 million by streamlining an arcane criminal justice system, they say the city could save at least $100 million a year by reducing the 12 times that the average defendant appears before a judge. The city hasn’t yet responded to that suggestion, says City Project executive director Bonnie Brower. But, she adds, “If he’s [the mayor] going to be recalcitrant on the revenue issues, he has to take a closer look at these substantive proposals being made by a variety of actors to make smart savings.”