I noted here some time ago that the next big battle in the war over women will be over the Paycheck Fairness Act, which would make it easier in various ways to challenge employers who engage in gender-based pay discrimination. Harry Reid has now promised a vote on the measure during the week of June 4th.

The bill probably won’t pass; it ran into a wall of GOP opposition in 2010. But I’m told Dems will move to make it an issue in multiple Senate races.

An aide to Senator Claire McCaskill tells me that her reelection campaign will make an “aggressive push” to get McCaskill’s GOP rivals, who are battling each other in a primary, to “explain why they’re opposed to paycheck fairness.” Two of the three have spoken out against federal wage-setting laws.

This will also figure in the battle between Elizabeth Warren and Scott Brown. A Massachusetts Democratic Party official tells me Dems will work to highlight Brown’s vote against the Paycheck Fairness Act in 2010 as proof that he doesn’t really have the middle class’s interests at heart. “We’re going to make the point that this isn’t just about women; it’s about middle class families, too,” the official says, adding that when women suffer pay discrimination, “two income households all get shortchanged.”

In Virginia, Dems are hitting GOP Senate candidate George Allen for refusing to take a position on the measure. In Nevada, Dem Senate candidate Shelley Berkley is criticizing Senator Dean Heller for voting against Paycheck Fairness and against the Lily Ledbetter Act, arguing: “It’s hard to believe that we’re talking about equal work and pay in 2012.”

With the Paycheck Fairness Act gaining a higher profile, the question will be how much longer Mitt Romney can avoid taking a position on it. The interesting wrinkle here is that Romney recently endorsed the concept of pay equity in general when he was questioned about Lily Ledbetter. But he hasn’t said whether he would have signed Lily Ledbetter, and he hasn’t taken a position on Paycheck Fairness, even though it will now be voted on in the Senate.

What’s particularly noteworthy about Romney’s reticence is that he has repeatedly argued that women won’t allow Obama and/or Dems to distract them from their economic woes by focusing on social issues, and that their economic circumstances will ultimately dictate their votes. But here is an issue that is all about women’s economic circumstances. It’s for this reason that Dems hope to highlight the GOP stance on Paycheck Fairness as a way to persuade women that Republicans are hostile to their economic interests — and to make the broader case that when it comes to women’s rights, the GOP remains trapped in the past in ways that hit them in the pocketbook — as part of their push to widen the gender gap for 2012.