A controversial law sharply curtailing collective-bargaining rights for Ohio’s public employees is sinking in the polls, raising Democratic hopes that the measure’s defeat could boost their prospects in the crucial swing state in 2012.

The law’s diminishing poll numbers have coincided with a decline in the approval ratings of Republican Gov. John R. Kasich, the measure’s most visible proponent. The drop is coming as the law’s union-led opponents have waged an energetic campaign against a measure that they say represents an overreach by the state’s Republican political leaders.

Now, analysts said, that effort could carry over into 2012, and national organizations are pouring millions into both sides of the referendum battle.

“The organizational efforts seem to be reviving portions of the Democratic coalition and that will likely have a beneficial effect on the presidential campaign for the Democrats,” said John C. Green, director of the Bliss Institute at the University of Akron.

The apparent unpopularity of the measure was highlighted this week when former Massachusetts Gov. Mitt Romney declined to endorse the law after visiting a Cincinnati call center where volunteers were working to build support for it.

But after being criticized for ducking the issue by conservative groups and fellow presidential aspirant Texas Gov. Rick Perry, Romney said Wednesday in Virginia that he was “110 percent” behind the measure.

Kasich, who was elected in a 2010 GOP surge, championed the law that eliminated many collective-bargaining rights for Ohio’s 350,000 public employees earlier this year. The legislative debate over the law drew thousands of protesters to the state Capitol in Columbus. Once the measure was enacted, opponents gathered 1.3 million signatures to put it on the Nov. 8 ballot.

Kasich said the law gives local governments in Ohio the tools they need to cut government spending, while ensuring that public employees pay for a fair share of their benefits.

He joined Republican governors in states including Wisconsin, Florida and New Jersey who sought to balance budgets by cutting government services and wresting deep concessions from public workers, in some cases by curbing the power of unions.

But some of those actions have fueled a political backlash.

In Wisconsin, Republican Gov. Scott Walker is facing a recall campaign that begins gathering signatures next month. That campaign came after union-led activists forced a series of recall elections that resulted in two GOP state senators losing their seats, leaving Republicans clinging to a one-vote majority in Wisconsin’s Senate.

Florida Gov. Rick Scott (R) has seen his approval ratings plummet after he implemented a series of deep budget cuts, championed corporate tax cuts and eliminated 15,000 public-sector jobs.

Meanwhile, 54 percent of Ohio voters disapprove of Kasich’s job performance, making him the most unpopular governor in the country, according to a recent survey by Public Policy Polling.

The law to limit collective bargaining is faring no better. A Quinnipiac poll this week found that 57 percent of Ohio voters support repeal of the law, while 32 percent oppose it. Opponents of the law have increased their lead since last month, when they held a 51-38 percent margin.

“The groundswell of enthusiasm and mobilization around this issue is something we have not seen in Ohio in a very long time,” said Seth Bringman, communications director for the Ohio Democratic Party. “There is no doubt that a year from now Ohio voters will remember who stood up for our middle class.”

Ohio Republicans dismissed the recent poll findings, arguing that the wording of the questions and the people actually canvassed skewed the results. Moreover, they say, the ballot question is unlikely to carry over into 2012.

“This is primarily a package of state reforms that help to empower local reform,” said Chris Maloney, communications director for the Ohio GOP. “I don’t think these reforms will have a tremendous amount of bearing on what happens in 2012.”

Still, Kasich is campaigning feverishly to keep the law in place. “We’re going to keep working,” he told reporters this week when asked about recent polls.“We’re making the case and we’re going to make it all the way up to Election Day, and we’ll see how it all comes out.”

The ballot question has attracted money from around the country. Supporters of the law argue that it is needed to help hard-pressed state and local governments get a grip on spending.

Building a Better Ohio, the group campaigning to keep the law, has spent nearly $6 million and has another $1.6 million on hand, according to its reports. That spending has been supplemented by a direct mail campaign and other help from outside organizations, including Alliance for America’s Future, a conservative group whose leadership includes Liz Cheney, the daughter of former vice president Dick Cheney.

In ads, advocates of the law assert that public employees make 43 percent more in pay and benefits than private-sector workers — a claim disputed by public-sector unions. “They want even more from us,” one ad says. “Better pay and benefits from us. Better job security than us. Better retirement than us. All paid for by us.”

Opponents have branded the law an attack on workers who form the backbone of the middle class, including police officers, firefighters and teachers.

We Are Ohio, the union-backed group pushing to repeal the law limiting collective bargaining , has spent more than $17 million and has another $4.3 million on hand, according to campaign finance reports filed Thursday.

“We have a lot of enthusiasm going for us. People are really engaged,” said Mark A. Sanders, president of the Ohio Association of Professional Fire Fighters.

Although voters seem to agree that the size and scope of government needs to be reduced in Ohio, there is also evidence that they think the measure has gone too far by attacking unions and some of the protections they offer workers.

The recent Quinnipiac poll found voters to be strongly supportive of parts of the law requiring public workers to contribute a minimum amount to their retirement and health-care costs. But voters strongly oppose other provisions in the law banning strikes, ending negotiations over health-care benefits and eliminating seniority rights, the poll found.

“Concessions are nothing new for public employees,” said Christopher Mabe, president of the Ohio Civil Service Employees Association. “We have been doing more with less for a long time because of Ohio’s weak economy.”

Combatants in the referendum battle said it is unclear what would happen if the law is defeated. Republicans have the option of reintroducing another, more limited, measure to limit collective bargaining in the next legislative session.

Meanwhile, if proponents of the law prevail, Republicans may be emboldened to go even further, analysts said.