WASHINGTON (Reuters) - Prominent Republican senators on Thursday embraced a push to overhaul rules for addressing sexual harassment in the U.S. Congress, signing on to a bill that would protect victims and require lawmakers to pay for their own settlements. The legislation builds on demands to lift the veil of secrecy around sexual harassment and misconduct on Capitol Hill, and has gained steam in recent months as a wave of women have come forward with accusations against prominent American men in politics, media and entertainment. The bipartisan push signaled momentum in the Republican-led U.S. Congress for overhauling a process for handling misconduct allegations that many lawmakers say is antiquated and stacked against victims. The Senate bill, called the Congressional Harassment Reform Act, draws from proposals that Senator Kirsten Gillibrand and Representative Jackie Speier, both Democrats, have been developing. “Congress is really behind the eight-ball. I think that, in many respects, the private sector has acted more swiftly than we have in terms of addressing sexual harassment,” Speier said in an interview. High-profile Republican senators co-sponsoring the bill include John Cornyn, the Senate’s No. 2 Republican; Ted Cruz; Joni Ernst and Lisa Murkowski. The legislation would require any member of Congress found liable for harassment to pay settlements themselves, rather than with taxpayer funds, as the current process allows. “Congress is not above the laws, and secret settlements with taxpayer money to cover up harassment should no longer be tolerated,” Cruz said in a statement. Settlements would be made public automatically unless victims choose to keep them private. Outrage over sexual misconduct in politics helped to fuel an upset victory by Democrat Doug Jones in the U.S. Senate race in deeply conservative Alabama on Tuesday. Voters rejected the Republican candidate in the race, Roy Moore, who had been accused by multiple women of pursuing them when they were teenagers and he was in his 30s, including one woman who said he tried to initiate sexual contact with her when she was 14. Moore denied the allegations but many prominent Republicans distanced themselves from Moore, although President Donald Trump backed him. In Washington, allegations of sexual misconduct prompted the resignations last week of three lawmakers - Democratic Senator Al Franken, Democratic Representative John Conyers and Republican Representative Trent Franks. On Tuesday, Republican Representative Blake Farenthold said he would not seek re-election in November. Politico reported that the congressional Office of Compliance had paid $84,000 from a public fund on behalf of Farenthold to settle a sexual harassment claim in 2015. Reuters has been unable to verify the allegations against Farenthold, who has said that the charges were false and has denied wrongdoing. The 1995 law governing the process for complaints in Congress - created in the wake of a harassment scandal - has been criticized as ineffective. The lengthy and cumbersome process requires victims to go through mandatory mediation and requires complete secrecy. “It created a protective blanket around the harasser and left the victim out in the cold,” Speier said. Speier, who has worked on the issue since 2014, came forward in October with her own story of unwanted sexual contact from the chief of staff for the lawmaker she worked for as a congressional aide. “He kissed me and stuck his tongue in my mouth,” said Speier, who has become a resource from women seeking advice on how to handle similar situations. “When it happened to me, it disgusted me. I kind of recoiled.” Speier’s proposals for reforms have attracted support from more than 100 members, including 19 Republicans. A group of conservative Republicans have championed a separate bill focused on banning the use of taxpayer dollars for settlements, and requiring past settlements to be disclosed and reimbursed. “What we do agree is that taxpayers should not be on the hook for misbehavior and for those settlements that are made,” said Marsha Blackburn, a Republican representative who has advocated for the proposal. “We need to use that to make certain that workplaces are respectful,” Blackburn said in an interview. A House committee is reviewing reforms with an eye to making recommendations in coming weeks. “I think that what we are doing is taking the best of all the ideas out there and putting them into one package,” a senior House Republican aide said.    