Republicans are deeply divided on the issue of same-sex marriage and, for the most part, see no point in discussing it. Party strategists say doing so only alienates younger voters.

But Republican officials scoffed at the notion that a ballot initiative in Ohio would help Democrats, predicting instead that a referendum would energize conservative opponents who would turn out to vote against it. “I think there’s great peril in them putting this on the ballot for their whole ticket,” said Matt Borges, executive director of the Ohio Republican Party. “You may awaken the sleeping giant.”

In Oregon, a federal court challenge to the state’s ban may make the ballot initiative moot. The Oregon attorney general, Ellen F. Rosenblum, announced Thursday that she would not defend the ban, and would instead argue that it “cannot withstand a federal constitutional challenge.” Backers of the ballot initiative said they would hold on to their signatures and await the outcome of the case. Oral arguments are scheduled for April.

“We’re pursuing both strategies,” said Mike Marshall, campaign manager for the group promoting the vote, Oregon United for Marriage. “We are totally ready to go on the ballot, but we’ve also got this great lawsuit that, like every other lawsuit, is looking more positive.”

Opponents of same-sex marriage are braced for a loss should the measure wind up on the ballot in Oregon. Brian S. Brown, president of the National Organization for Marriage, said that much of his group’s work this year would be focused on defeating candidates in Republican primaries who support same-sex marriage. In Ohio, he is looking ahead to 2016, when Senator Rob Portman, the Republican who reversed his position on same-sex marriage after his son came out as gay, will be up for re-election.

Proponents of same-sex marriage are also looking to 2016, when they say voter turnout will be more favorable during a presidential race. Freedom to Marry is laying the groundwork for votes in Arizona, Colorado, Michigan and Nevada — and Ohio.

But the two-year-old Freedom Ohio does not want to wait. Mr. James, the executive director, said the group had gathered far more than the 386,000 signatures needed to put a measure on the ballot. It will make a final decision closer to the July 2 filing deadline, he said, taking fund-raising into consideration.