For opponents of same-sex marriage, it has been a potent and constantly repeated talking point: Though the courts or the legislatures of some states have given gay and lesbian couples the right to marry, wherever it has appeared on the ballot, voters have rejected it.

Gay rights advocates hope that after Tuesday’s election, no one will be able to say that any more.

With public opinion shifting in their direction, a 4-to-1 spending advantage and months or even years of one-on-one appeals to potential voters, rights groups see a good chance of victory in at least one or two of the four states where same-sex marriage is a ballot item this year.

Current polls indicate a solid lead for supporters in Maine and a lesser one in Washington state, while the races in Maryland and Minnesota are about even, with the opponents apparently gaining.

Even a single victory "will be a turning point," said Chad Griffin, president of the Human Rights Campaign, a national group promoting gay rights that has directed $5 million to the four marriage battles.

Six states and the nation’s capital have legalized marriage for gay and lesbian couples. But in 30 states, in constitutional amendments, voters have limited marriage to a man and a woman; in addition, same-sex marriage has been blocked in referendums like those in California in 2008 and Maine in 2009.

Opponents of broader rights for gay and lesbian couples are mounting a final-week barrage of advertising and telephone appeals, warning undecided voters that "redefining marriage" would force onerous changes in schools, businesses and churches. Rights groups denounce those messages, which are similar to those broadcast in previous ballot fights, as misleading scare tactics.

"Our ads have started in earnest, and they will be airing more heavily in the final week," said Frank Schubert, a California-based consultant who is managing all four state campaigns against same-sex marriage, mainly with financial aid from the National Organization for Marriage and affiliates of the Roman Catholic Church.

For weeks, supporters of same-sex marriage have been showing advertisements of their own in which community members say that gay and lesbian friends deserve the same chance to love and marry that everyone else enjoys.

Legislatures in Washington and Maryland approved same-sex marriage earlier this year, but opponents gathered signatures to force public referendums on the laws.

It has been three years since voters repealed Maine’s same-sex marriage law. Gay rights supporters have spent these years pursuing conversations with potential voters and appear to have persuaded enough of them to win, political experts said.

Minnesota already has a law barring same-sex marriage, but conservatives are trying to pass an amendment to the state constitution banning it as well, to prevent a new legislature from reversing the law. Despite being outspent nearly threefold, they appear to have the upper hand.

Perhaps the most expensive campaign is in Washington state, where supporters of Referendum 74, to endorse same-sex marriage, have raised nearly $11 million, including $2.5 million from Jeff Bezos, the founder of Amazon.com, and his wife, Mackenzie, and $600,000 from Bill and Melinda Gates.

The opponents have raised $2.4 million, including $1.1 million provided by the National Organization for Marriage.

Matt Barreto, a political scientist and pollster at the University of Washington, said that polls showing a 10-point advantage for the referendum may be misleading, because some voters listed as undecided are embarrassed to voice opposition. But he said that a steady 53 percent of Washington voters express support, and he expects the referendum to be approved.

One of the phone bank volunteers for Washington United for Marriage, Damian Murphy, 49, described a call in which he tried to reassure a man with unjustified fears. The man worried that his pastor would be convicted of a hate crime if the law passed; Murphy read the measure aloud, including assurances that religious organizations could not be sued for refusing to marry gay couples.

The man said, "Thank you, I’ll think about it," Murphy recalled.

Schubert, the manager of the campaigns against gay marriage, agreed that the effort was trailing in Washington but said the gap was narrowing. Schubert is known as the mastermind of the come-from-behind victory of Proposition 8 limiting marriage to a man and a woman in California in 2008, and he sought to play down the idea that a victory for same-sex marriage in one or two states would be a milestone.

Maine, he said, will become one more state in a liberal corner of the country to permit same-sex marriage. Washington is also a liberal state, and "it would be unfortunate if we lose there, but not very surprising," he said, given the disparity in fundraising.

On the other hand, he said, if his opponents lose in Washington despite a 5-to-1 spending advantage, they will have to think twice about promoting new ballot initiatives in less liberal states, like California and Oregon.

Griffin of the Human Rights Campaign said he expected Tuesday’s elections to signify a historic shift.

"The struggle for equality will by no means be finished, but it will be seen as a turning point as we begin to win more and more of these votes," Griffin predicted.

© 2012 The New York Times Company