WASHINGTON — Lawmakers are facing mounting pressure to end Capitol Hill’s culture of secrecy over sexual harassment as they return from a holiday break, with members of both parties calling for Congress to overhaul its handling of misconduct claims and to unmask lawmakers who have paid settlements using taxpayer money.

On Sunday, the roiling debate over sexual harassment cost one lawmaker who has paid such a settlement — Representative John Conyers Jr. of Michigan — his post as the top Democrat on the House Judiciary Committee, at least temporarily. Mr. Conyers, the longest-serving member of the House, announced that he was stepping aside as the House Ethics Committee investigates allegations that he sexually harassed aides.

And on the other side of the Capitol, Senator Al Franken, the Minnesota Democrat who has been accused of groping several women, told a home state newspaper that he would return to work on Monday feeling “embarrassed and ashamed.”

The announcements by Mr. Conyers and Mr. Franken came as both Democrats and Republicans took to the Sunday morning television talk shows to call for greater transparency in how harassment claims are dealt with. Under a 1995 law, complaints are handled confidentially. Lawyers for the House and the Senate have required that settlements be kept confidential as well.

“All of this, as difficult as it is in some respects for our society, is really important because I think it will end up changing people’s attitudes and changing our culture,” Senator Rob Portman, Republican of Ohio, said on NBC’s “Meet the Press.” “So I am glad it’s being discussed. I think it should be more transparent. I certainly think that if you accept taxpayer funds for settlement, that should be transparent.”

The House is expected this week to adopt a bipartisan resolution mandating that all members and their staffs participate in anti-harassment and anti-discrimination training; the Senate has already adopted such a resolution. The more difficult task will be passing legislation that overhauls the way sexual harassment claims are handled.

In the House, a bipartisan group of lawmakers, led by Representative Jackie Speier, Democrat of California, and Representative Barbara Comstock, Republican of Virginia, is pushing for legislation that would require claims to be handled in public. In the Senate, Senator Kirsten Gillibrand, Democrat of New York, has put forth similar legislation.

“It was a system set up in 1995 to protect the harasser,” Ms. Speier said on the ABC program “This Week,” adding, “We say zero tolerance, but I don’t believe that we put our money where our mouths are.”

One major question, however, is whether the Speier-Comstock legislation should apply retroactively, meaning that those who have paid past settlements would now be identified. The legislation would cover any settlement reached since the beginning of this year.

While Mr. Portman said he would support retroactive releases, others, including Representative Nancy Pelosi, the House Democratic leader, were more cautious, saying that unmasking lawmakers could reveal the identity of victims who want to remain private.

“All of these nondisclosure agreements have to go,” Ms. Pelosi said on “Meet the Press.” But, she said, “if the victim wants to be private, she can be.”

Debra Katz, a lawyer who represents victims of sexual harassment, echoed those concerns.

“For a number of my clients, that’s the last thing in the world they would want and could have life-altering consequences,” Ms. Katz said in an interview on Sunday. “They settled their cases to be able to move on with their lives while protecting their privacy.”

In the case of Mr. Conyers, the lawyer Lisa Bloom, who announced on Sunday that she was representing the woman who filed the complaint against him, said a confidentiality agreement was preventing the woman from telling her side of the story. Ms. Bloom urged Mr. Conyers to release her client from the agreement so she could speak publicly.

News of Mr. Conyers’s settlement was reported last week by BuzzFeed News, which published documents showing that he had settled a complaint in 2015 by a former employee who had said she was fired because she rejected his sexual advances. The news site said it had received documents about the case from Mike Cernovich, a right-wing online commentator.

BuzzFeed has since reported that a second woman has also accused Mr. Conyers, 88, of sexual harassment.

“I deny these allegations, many of which were raised by documents reportedly paid for by a partisan alt-right blogger,” Mr. Conyers said in a statement on Sunday. “I very much look forward to vindicating myself and my family before the House Committee on Ethics.”

Mr. Conyers said that he would “like very much to remain as ranking member,” but had “come to believe that my presence as ranking member on the committee would not serve these efforts while the Ethics Committee investigation is pending.”

His lawyer, Arnold E. Reed, said in a phone interview on Sunday that Mr. Conyers had taken several days to decide to step aside from his committee post because he did not want to make an “off the cuff” move. Mr. Conyers spoke with several family members and deliberated during the Thanksgiving holiday before determining that the allegations had become too much of a distraction, the lawyer said.

“He wanted time to think about this and reach a conclusion that he was comfortable with. And it was the right thing to do in his mind,” Mr. Reed said. “He is maintaining that he did not do anything wrong. He is maintaining his innocence. This is a temporary stepping aside his position as ranking member so this can be a completely transparent and unfettered investigation.”

On Wednesday, Mr. Reed had said in an interview that Mr. Conyers believed that some of those suggesting that he step down, including fellow Democrats, had been scheming for years to push him out of his Judiciary post.

A senior House Democratic aide said the decision had come after days of effort by Ms. Pelosi, who was working with Mr. Conyers to find a way for him to step aside gracefully. Ms. Pelosi hinted at as much on “Meet the Press,” where she said, before Mr. Conyers’s announcement, that she expected him to “do the right thing.”

The interview showed the delicate position that Ms. Pelosi is in. She declined to say that Mr. Conyers should step down, calling him an “icon in our country” who had done “a great deal to protect women.” Ms. Pelosi later came under some criticism on social media for those remarks.

On Sunday night, 12 women who once worked for Mr. Conyers released a statement in support of him. “Our experiences with Mr. Conyers were quite different than the image of him being portrayed in the media,” the women said, adding that he was “respectful” and “treated us as professionals.”

Former Franken aides have also been coordinating an effort to line up women in support of him. On Sunday, they released a statement signed by 65 women that expressed disappointment over the allegations but called him a “steadfast supporter of women’s rights.”

Representative Jamie Raskin, a Maryland Democrat who holds the recently created position of vice ranking member on the Judiciary Committee, praised Mr. Conyers for making a “wise decision,” adding, “The House is ready to clean house with respect to sexual harassment, and everybody agrees that we need to have a zero-tolerance policy.”

As Democrats wrestled with the allegations against Mr. Conyers and Mr. Franken, congressional Republicans on Sunday bemoaned President Trump’s support for Roy S. Moore, the Republican Senate candidate in Alabama who is accused of making unwanted advances on teenagers.

Many Republicans on Capitol Hill have called for Mr. Moore to step aside, but he has refused to do so. In a pair of tweets on Sunday, Mr. Trump warned that electing Mr. Moore’s Democratic opponent, Doug Jones, “would be a disaster!” Senator John Thune, Republican of South Dakota, issued his own warning, saying that a victory by Mr. Moore would hurt Republicans just as much as a loss.

“If Moore wins, there will immediately be an ethics investigation, and he will be working under a cloud. He is a distraction,” Mr. Thune, the No. 3 Republican in the Senate, said on “Fox News Sunday.” “I would like to see the president come out and do what we’ve done, saying Moore should step aside.”