Some conservatives raise an eyebrow over unions’ claims that they are outgunned in the money game. “I look at them saying they’re going to be outspent, and I say, I don’t know, maybe that’s the case, but it also sounds an awful lot to me like they’re trying to manage expectations,” said Seth Morgan, the Ohio director of policy for Americans for Prosperity. The group says it is opening seven offices in the state and has called more than one million Ohio voters. “Think about it: they’ve got a built-in funding mechanism after all,” Mr. Morgan said.

If labor cannot provide the counterpunch to the conservative super PACs, it is unclear whether anyone else can. Nationally, organized labor has long been viewed as having the most effective political operation for Democrats. President Obama’s victory in 2008 here in Ohio — no Republican in modern times has been able to capture the White House without winning the state — was due in no small part to labor’s get-out-the-vote push. People from union households represented 30 percent of all who voted in the state that election.

But over the past two years, unions have been diverting resources to a range of causes beyond presidential politics, including contract showdowns with companies like Caterpillar and legislative battles over a wave of state laws diminishing collective bargaining power. Still, union leaders argue that the battles over public employees’ bargaining rights in Wisconsin and elsewhere brought new momentum and vigor to union activists, who pushed successfully to repeal a Republican-backed anti-bargaining law in Ohio in a statewide referendum last November.

“What we saw in the past two years in all these state battles across the country is members coming out of the woodwork to volunteer,” said Lee Saunders, the president of the American Federation of State, County and Municipal Employees.

Moreover, the same Supreme Court decision that permitted unlimited donations to super PACs has also in ways benefited unions. The ruling in the Citizens United case and subsequent court rulings opened the door to unlimited corporate and union contributions to political committees and made it possible to pool that money with unlimited contributions from wealthy individuals. It also allowed unions to campaign beyond their membership for the first time — to call and knock on the doors of nonunion households. As a result, unions boast that they will reach a far larger universe of voters than ever in 2012.

Here in Ohio, union leaders expect that a record number of volunteers, pressing for Mr. Obama and Senator Sherrod Brown, a Democrat and a favorite of labor, will call or knock on the doors of 2.3 million voters by Election Day, Nov. 6, double the number they reached in the last presidential race.