Senators are nearing an agreement on changes to the chamber’s sexual harassment policy, with key lawmakers saying on Monday a deal could be introduced this week.

"We are very close and I still think there's a chance we'll get something done in the Senate before we leave this week. That's how close I think we are," Sen. Roy Blunt Roy Dean BluntGOP, Dem lawmakers come together for McCain documentary The Hill's 12:30 Report The Hill's 12:30 Report MORE (R-Mo.), the chairman of the Rules and Administration Committee, told reporters.

Sen. Amy Klobuchar Amy Jean KlobucharHillicon Valley: Lawmakers target Chinese tech giants | Dems move to save top cyber post | Trump gets a new CIA chief | Ryan delays election security briefing | Twitter CEO meets lawmakers Twitter CEO meets with lawmakers to talk net neutrality, privacy GOP, Dem lawmakers come together for McCain documentary MORE (D-Minn.), who has been leading negotiations for Democrats, added that she also hopes to have a deal to announce before the Memorial Day recess and that "it's going very well."

Senators were tightlipped about any details of the forthcoming legislation and a Senate aide noted separately that bipartisan talks are still ongoing.

"I think everybody is generally in agreement on a fairly specific concept but we're waiting to get the exact language back from the drafting people," Blunt said when asked if Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellTrump's plan to claw back spending hits wall in Congress GOP lawmakers want Trump to stop bashing Congress Parkland father calls out Trump, McConnell, Ryan after Santa Fe shooting MORE (R-Ky.) has signed off on the changes.

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A spokesman for the GOP leader said on Monday evening that senators are continuing to work on the legislation and they didn't have any announcements.

The majority leader's support would be crucial to getting floor time and a vote on any legislation.

A spokesman for Senate Minority Leader Charles Schumer Charles (Chuck) Ellis SchumerSchumer: GOP efforts to identify FBI informant 'close to crossing a legal line' Patients deserve the 'right to try' How the embassy move widens the partisan divide over Israel MORE (D-N.Y.) didn't immediately respond to a request for comment.

But Klobuchar noted that the Senate Democratic leader had also been involved in the talks, and Schumer has previously called for updating how sexual harassment claims are handled on Capitol Hill.

A bipartisan group of senators, led by Blunt and Klobuchar, have been privately negotiating for months on legislation to overhaul the Senate's handling of sexual harassment claims.

"I'm working with a bipartisan group of senators right now who are trying to negotiate a final resolution so we can get a bill on the floor and vote on it hopefully as early as next week," Democratic Sen. Kirsten Gillibrand Kirsten Elizabeth GillibrandOvernight Health Care — Sponsored by PCMA — Trump hits federally funded clinics with new abortion restrictions Dem senators ask drug companies to list prices in ads Gillibrand to publish children's book about suffragists MORE (N.Y.) told MSNBC in an interview that aired on Sunday.

Blunt noted that while he expects any legislation would skip the committee process and go straight to the floor, Senate passage this week "might be a little optimistic."

Senators on Monday declined to discuss in detail how their forthcoming legislation would be different from legislation passed by the House or how it would address the issue of personal liability, viewed as a thorny issue amid reports that some senators had concerns about making changes to who pays for sexual harassment settlements involving lawmakers.

"We're very close to [being] ready to talk about this and I think it would be better for us all to be talking about it at the same time, OK?” Blunt told reporters, when asked if they had come to an agreement on the provision.

Klobuchar told The Hill that individual liability will be addressed to some extent.

"There will be some individual liability, trust me. It will be there," she added. "Wait till you see what we got. I worked hard on it."

Currently, settlements involving lawmakers are paid by using taxpayer funding from a little-known Treasury account. The Washington Post reported in December that the Treasury has spent $174,000 on settling harassment-related claims over the last five years.

A House bill passed in February would reform the Congressional Accountability Act and put lawmakers on the hook for any settlement payments.

Senators have been under growing pressure from their own colleagues to pass legislation reforming how the chamber handles sexual harassment complaints.

Every female senator sent a letter to leadership in March arguing that the current system, set up by the Congressional Accountability Act, is "antiquated."

"No longer can we allow the perpetrators of these crimes to hide behind a 23-year-old law. It’s time to rewrite the Congressional Accountability Act and update the process through which survivors seek justice," they wrote.

A spokesperson for McConnell told NBC News in wake of the letter that the Kentucky Republican "supports members being personally financially liable for sexual misconduct in which they have engaged."

Most male Democratic senators, joined by GOP Sen. Ted Cruz Rafael (Ted) Edward CruzTed Cruz and Bill Nelson give NASA a reality check on privatizing International Space Station Ten dead after shooting at Texas high school Hillicon Valley: Lawmakers target Chinese tech giants | Dems move to save top cyber post | Trump gets a new CIA chief | Ryan delays election security briefing | Twitter CEO meets lawmakers MORE (Texas), sent a similar letter in April.

Legislation was left out of a March spending bill but CNN reported earlier this month that senators were picking up the pace on negotiations.

The Senate separately cleared a resolution late last year requiring that all senators and staffers undergo sexual harassment training.