In contrast to Wisconsin’s Republican governor, who has targeted public-worker unions as the chief villain of his state’s budget-cutting drama, Democratic governors across the country who face similar fiscal challenges have tried to sidestep such confrontations with a key constituency by quietly cutting deals with labor leaders.

California Gov. Jerry Brown has proposed largely sparing schools and prisons from the deeper cuts hitting other areas as he tries to close the state’s $25 billion deficit, and powerful unions representing teachers and corrections officers that poured millions into helping elect Brown last year are now lining up behind his budget plan.

New York Gov. Andrew Cuomo, who has angered public-sector unions by calling for deep reductions in benefits, has worked closely with some labor officials on proposed cutbacks and is promising more dialogue in the coming days.

And while Colorado Gov. John Hickenlooper recently proposed deep cuts to Medicaid, human services and education and even called for closing a state prison, he has vowed to retain an agreement giving workers limited bargaining rights and has invited employees to submit ideas for cutting waste and inefficiencies.

“I’m not saying the unions are happy about this,” said Maryland Gov. Martin O’Malley, who has angered public-worker advocates by his push to overhaul state pensions as he tries to cut billions from the budget this year. “But it’s not like we’re locking each other out of the statehouse or stopped talking with each other.”

O’Malley, chairman of the Democratic Governors Association, will welcome a number of his counterparts from around the nation to Washington for a weekend meeting, starting Friday, at which budget-cutting strategies are expected to be a hot topic. He described his party’s approach with the unions as “fundamentally pragmatic,” adding that Wisconsin Gov. Scott Walker and other Republican leaders have embarked on an “ideological drive to go after the unions, to destroy the unions.”

Walker’s plan to roll back collective-bargaining rights for many public workers has sparked mass protests and a legislative standoff. It has also suddenly made him a hero to many conservatives who see public-sector unions as major funders of Democratic campaigns and fierce defenders of pension systems that have helped drain state coffers.

Walker has said that his sole motive is to regain control of Wisconsin’s budget and that unions’ collective-bargaining powers unduly constrain state and local governments. The governor’s plan “is about avoiding massive layoffs and balancing this and future budgets,” said spokesman Chris Schrimpf. “That’s what he was elected to do.”

The decision by Walker and some other new Republican governors, such as Ohio’s John Kasich, to take on public-worker unions carries some risk. For example, their efforts have mobilized liberal activists in two states likely to be critical battlegrounds in President Obama’s 2012 reelection campaign.

But Democrats from Obama to city mayors face a larger political challenge: how to satisfy the public pressure — and in some cases, legal obligation — to balance budgets through spending cuts without antagonizing or alienating some of their most loyal electoral and financial supporters.

The White House has tried to find the right balance. Obama last week called Walker’s plan an “assault” on unions, and the president’s political organization helped instigate demonstrations in Madison, Wis., and other state capitals. But Obama and his aides have also emphasized his support for less federal spending, including a pay freeze for federal workers.

The tension is palpable even in some deeply Democratic cities and states, where unions and politicians have enjoyed fruitful alliances for years. Now voter anxiety over government deficits is putting more distance between the two groups.

In Chicago, former White House chief of staff Rahm Emanuel won an easy majority in the city’s six-way mayoral race on Tuesday — even though he pledged deep cuts to city pensions and did not have the support of powerful police and firefighter unions. Emanuel has pledged to work with the unions when he takes office, but his decisive victory gives him a mandate to push for cutbacks.

Brown, perhaps more than other Democrats, has sought a full partnership with public-sector unions. He has proposed slicing $12.5 billion from the state budget, with deep cuts to universities and human services and pay reductions for state workers, but unions like his plan to ask voters to approve in June $12.5 billion in temporary taxes. Public-sector unions, such as the powerful California Teachers Association, have signaled they would fund the media campaign in support of passage for the taxes.

If state lawmakers don’t agree to the referendum or if voters reject the tax extensions, Brown warned Thursday, he would push for the full $25 billion in budget cuts.

“It’s crafty,” said Joe Mathews, co-author of the new book “California Crackup: How Reform Broke the Golden State and How We Can Fix It.” Brown is “making the unions in some sense own this budget proposal. If it doesn’t work, he can say, ‘I tried. Now we’ve got to do this hard stuff.’ ”

Cuomo has reached out to New York’s most powerful union, Service Employees International Union 1199, which represents thousands of health care workers but not public employees. He solicited the group’s participation on a panel studying Medicaid cuts that could affect many of the union’s 275,000 members in the state.

But in his campaign last year, Cuomo called for sacrifice by public-sector unions, penning a Labor Day column for the New York Daily News titled, “Labor, be part of the solution.” And some public-employee advocates say they remain unsure what to make of the new governor, who has invited them to negotiate over hundreds of millions of dollars in proposed cuts to payroll and benefits but hasn’t worked yet on specifics.

One union official said Thursday that his members are giving Cuomo the benefit of the doubt, but labor is bulking up its social-networking abilities in case more grass-roots action is needed.

“We’re baffled here because he’s saying he wants to negotiate with us but we haven’t even sat down and talked about it,” said Stephen Madarasz, spokesman for the Civil Service Employees Association, one of New York’s public-worker unions. “So we have no idea how he’s looking to get that savings.”

Hickenlooper, a self-described centrist Democrat, said Thursday that he has tried to enlist groups as disparate as public-worker unions and Republicans in helping figure out the state’s budget dilemma — an effort aimed at building bridges with key constituencies, rather than burning them.

“We’re going to go through a very hard time, and we need everybody’s help,” he said. “As much as we can, we want to turn down the emotion. If you have to make bitter choices, how can we help each other to do this?”

Some of Hickenlooper’s recent proposals would cause more pain to state workers, who are heading into their third year without a pay raise. For instance, he plans to close at least one state park as well as a prison in southeastern Colorado and to ask workers to pay more than they have toward their pensions.

Teacher unions were not pleased with the deep cuts that Hickenlooper has proposed to education funding in the state.

“Nobody is happy about that,” Hickenlooper said Thursday.

But despite that discontent, he said, the unions have been willing to work with him on where the cuts should come.

“I think they recognize we don’t have a whole lot of choice,” Hickenlooper said.

Asked about Walker’s tactics in Wisconsin, the Colorado governor demurred.

“I’m not great at criticizing someone else,” he said, noting that almost every governor these days has his own unique and perplexing set of problems.