As the House voted this week on the Respect for Marriage Act, which codifies federal protections for same-sex couples that were put in place in a 2015 ruling, 47 Republicans voted “yes.” That raised the possibility that the legislation could make its way to President Biden’s desk for his signature.

Minority leader Mitch McConnell, a Republican from Kentucky and obstacle to most Democrats’ agendas, declined to comment. On Wednesday, Susan Collins of Maine, Rob Portman of Ohio, Lisa Murkowski of Alaska, and Thom Tillis of North Carolina supported it.

Senator Chuck Schumer of New York, the majority leader, switched from declining to commit to a vote on Tuesday. Additionally, Mr. Schumer said he was working on getting the 10 Senate Republicans needed to overcome the filibuster.

This legislation received bipartisan support in the House, he said.

There was no guarantee that Republicans would support the measure in the Senate. After the midterm elections, it became clear that the evenly divided Senate would weigh in on whether same-sex couples should be allowed to marry – a bill that was supposed to die.

Mr. Schumer said Senator Tammy Baldwin, the first openly gay woman elected to the Senate in 2012, spoke with Republicans to see if the bill would pass. Aides said he discussed it with Ms. Collins, a co-sponsor.

The push in Congress to pass legislation codifying marriage protections came after Justice Clarence Thomas suggested last month, in an opinion overturning abortion rights, that the court “should reconsider” past rulings establishing marriage equality and access to contraception.

With their control of Congress hanging in the balance in November’s elections, Democrats are seeking to draw clear distinctions with Republicans on issues that have broad resonance for the public. The overturning of Roe v. Wade last month dramatized the stakes and drove home the prospect that the court could strip away more protections with no recourse in Congress should Republicans win the majority.

The demise of Roe also prompted outrage among progressives who had harshly criticized Democratic leaders for having failed to safeguard abortion rights when they had the chance and being slow to respond to a Supreme Court ruling that had been expected for months.

But on the issue of same-sex marriage, as opposed to abortion, Republicans are deeply divided. Many conservative lawmakers have switched positions over the past decade as the country has come to accept same-sex marriage as a settled matter.

About 71 percent of Americans, including most Republicans, support it, according to a recent Gallup poll, up from just 27 percent in 1996.

Mr. Portman, co-sponsoring the legislation, flipped his position in 2013 after his son came out as gay. In the House, Representative Liz Cheney, Republican of Wyoming, admitted last year that “I was wrong” to oppose same-sex marriage, reversing a longstanding position that had put her at odds with her family, including her sister, who is gay and married.

Still, while the proportion of House Republicans who supported the marriage equality legislation this week was higher than expected, it was less than one-quarter of the conference. Most Republican senators were similarly unenthusiastic about the bill.

On Wednesday, many were either dismissive or evasive about how they would vote, with some accusing Democrats of trying to distract from inflation and other pressing national issues. Others contended that the vote was unimportant or unnecessary because same-sex marriage protections were not under genuine threat.

Senator John Thune, Republican of South Dakota, told CNN that he respected the 2015 Supreme Court decision in Obergefell v. Hodges that legalized same-sex marriage nationwide but that overturning it was not “an issue right now that anybody’s talking about.”

Justice Thomas did talk about it in his recent opinion, and Justice Samuel A. Alito Jr. has also suggested before that Obergefell should be revisited, arguing that it invented a right with no basis in the text of the Constitution.

Senator Tommy Tuberville, Republican of Alabama, said he saw no need to pass legislation to protect gay marriage but supported it in practice.

“Yeah, if that’s what you want to do, fine,” he told reporters.

Senator Marco Rubio, Republican of Florida, said he would oppose the bill and told CNN that it was a “stupid waste of time.”

He added, “I know plenty of gay people in Florida pissed off about gas prices.”

Senator Mitt Romney, Republican of Utah, said he was not focused on the bill because Obergefell still protected same-sex marriage.

“I don’t think we need to lose sleep over it unless there was a development that suggested the law was going to be changed,” Mr. Romney said.

Senate Democratic aides said it was encouraging that House Republican leaders did not whip “no” votes on the bill on Tuesday, indicating that the party was divided on the issue of same-sex marriage. The party’s leaders split on the account.

The top two Republicans, Representatives Kevin McCarthy of California and Steve Scalise of Louisiana, voted no. But the No. 3 Republican, Representative Elise Stefanik of New York, and Representative Tom Emmer of Minnesota, the G.O.P. campaign committee chairman, were in favor.

Still, it was unclear where six more Republican votes in favor of the legislation could be found in the Senate.

Mr. McConnell, who has been explicit in the past about the party’s need to appeal to suburban voters, was not tipping his hand.

“I’m going to delay announcing anything on that issue until we see what the majority leader wants to put on the floor,” he told reporters on Tuesday.

The bill, which codifies the right to marriage regardless of gender or race, also protects interracial marriage. Mr. McConnell’s wife, Elaine Chao, was born in Taiwan.

Democrats saw no political downside in waiting to see if they could muster enough votes for passage. Until then, they noted privately that they were forcing Republicans to squirm as reporters peppered them with questions about an issue that most of the country had long since decided was uncontroversial.