Mitt Romney routinely accuses Obama of failing to get tough with China, and the President has returned fire by slamming Romney over Bain outsourcing and touting his own filing of enforcement actions against the country.

But it’s worth noting that there is a China-currency bill sitting right there in Congress that has already passed the Senate and would almost certainly pass the House if the GOP leadership scheduled a vote on it.

That bill is now going to get some more attention and could roil the politics a bit around China in the presidential race. Although the bill has strong support in Congress, in both parties, the White House has not publicly supported it, making this a point of disagreement with labor, which views it as a major priority. Romney opposes it.

Senator Sherrod Brown, the lead sponsor, is organizing to increase the pressure on his fellow Ohioan, John Boehner, to hold a vote on the bill. He is circulating a letter today among fellow Senators, addressed to Boehner, demanding a vote, and noting that a similar version passed in 2010 with the support of 80 House Republicans still in office.

The bill would give the U.S. government trade tools to more easily combat currency ma­nipu­la­tion, which could lead to higher tariffs against China. Steven Dennis has a good piece on the backstory and the politics here; the short version is that the White House has not supported it because it believes this course of action would start a trade war with China. But many groups on the left want it to pass as part of their push to revitalize manufacturing.

In an interview with me, however, Senator Brown told me that he’s convinced Obama would sign the bill if it passed — a declaration that will catch the attention of the major unions who want it.

“I’m confident Obama will sign this bill,” Brown said, adding that his conversations with the White House had persuaded him of this. “This will pass if Boehner schedules it.”

Brown added that the bill, and the ideas behind it, resonate heavily with working class voters, such as those in industrial swing states like Ohio and Wisconsin. “People recognize that trade policy and tax policy have undermined the middle class and manufacturing,” Brown said. “The public knows the game has been rigged.”

Brown’s declaration that Obama would pass the bill seems intended to increase pressure on House GOP leaders to bring it to a vote; China is a potent issue in the industrial battlegrounds. But it seems highly unlikely that it will come to a vote, and the White House will probably continue to keep its position on the issue vague. Still, renewed attention to the bill could perhaps increase the pressure on all parties to act.