By Jessica Dean, CNN

(CNN) -- A bipartisan and bicameral group of congressional negotiators met again Tuesday as they try to craft a compromise bill overhauling the nation's policing laws, with one week left until the one-year anniversary of George Floyd's death.

The lawmakers agree they are not tied to a deadline, despite President Joe Biden calling for a bill by the anniversary on May 25. After Thursday, the House does not have votes scheduled for the rest of the month, making passing any legislation by next week all but impossible.

"The most important thing is that we have a bill that hits the President's desk, not the date that it does," Rep. Karen Bass, a California Democrat, told reporters following Tuesday's meeting.

The meeting included members of the House Problem Solvers Caucus, Democratic Rep. Josh Gottheimer of New Jersey and Republican Rep. Brian Fitzpatrick of Pennsylvania, in addition to Bass, Democratic Sen. Cory Booker of New Jersey and Republican Sen. Tim Scott of South Carolina. The participation of additional House members is an indicator of the need for any deal to have bipartisan, bicameral support.

"Every meeting we have is just good energy ... sincere, honest talk and that just gives me encouragement, but we still have a hill to climb," Booker said.

Scott, who is leading negotiations for Senate Republicans, said, "The key for us is to keep making progress."

"I've been saying that constantly because really negotiating with anybody other than those folks who have votes doesn't really make a lot of sense to me. But I will say this is another constructive meeting and we continue to make progress. And in my opinion, that is the goal is to serve the communities that are the most fragile and often times the most vulnerable and the officers that have to come into those communities. By doing both, I think we solve a lot of problems."

Asked if the group has made any progress on one of the key sticking points, Section 242, the federal law that sets the standard for criminally prosecuting police, Booker said, "we are working hard on making sure there's real accountability for police officers, both civil and criminal. And that's my goal, it was in the Justice in Policing Act, but this is a negotiation. This bill can't get done if it doesn't have Senate Republican support, not just some support, it's really got to be like the First Step Act where you see a significant amount of senators on both sides supporting it."

Some progressive Democrats have balked at the idea of any compromise on key issues like Section 242 and qualified immunity. But Booker said this bill will not be a "cure-all."

"This bill in no way is going to address all the challenges we have around race and public safety. It's just not. So to set that expectation is ... wrong," Booker said. "What we are trying to do on a very long journey in America, is to take good strides forward and make meaningful progress to have substantive reforms that will make a difference in creating more safety and creating more progress."

"It's just very, very important that this bill not be seen as something that's a cure-all for something but something that I hope, if we can get it done, will be substantive and meaningful progress, significant progress," he added.