By Kristina Peterson, Andrew Duehren, Wall Street Journal–

Congressional Republicans are crafting their own proposals to respond to the killing of George Floyd while he was in police custody and the ensuing nationwide protests, marking a shift in the party’s stance on overseeing police practices and raising the prospect of collaboration with Democrats.

In the sharply divided Congress, the uproar over Mr. Floyd’s death and the underlying issues it exposed has spurred lawmakers to sift through many ideas aimed at curbing abuses by police officers. Mr. Floyd, a 46-year-old black man, was killed after a white police officer placed his knee on his neck for nearly nine minutes.

The Republicans’ plan will respond to the “obvious racial discrimination that we have seen on full display on our television screens over the last two weeks,” said Senate Majority Leader Mitch McConnell (R., Ky.). The early support for legislative changes could quickly fizzle in an election year or be derailed by policy debates. But Republicans said there was a new willingness to engage on the issue, and recent polls have shown broad public concern over police treatment of black Americans.

“Both parties are responding to the moment and to the voices of strong majorities of Americans in both parties who are saying that this needs to be looked at,” said former Senate GOP leadership aide Brian Walsh.

The Senate’s only black Republican, Sen. Tim Scott of South Carolina, on Tuesday walked his colleagues through a plan he is working on with a handful of other senators. It would collect more data from police departments about when force is used, increase the use of body cameras, implement new training to de-escalate tense situations and potentially other measures. Mr. Scott said he hoped to complete his package by the end of the week.

“It’s important for this nation to take a very powerful stand and position that says we’re listening, we’re hearing and we’re reacting,” Mr. Scott told reporters after the Senate GOP’s weekly lunch. “We’re responding in a positive, constructive manner that doesn’t create a binary choice between supporting law enforcement and supporting communities of color, and I think you can actually do both.”

Mr. Scott also met Tuesday with White House chief of staff Mark Meadows and other officials.

“We’re hopeful that we can address the issue in a real way,” Mr. Meadows told reporters Tuesday after meeting with Mr. Scott, saying the administration was “hopeful for something sooner than later.”

The discussions came on the day Mr. Floyd was being laid to rest in Houston.

President Trump has repeatedly emphasized “law and order” in response to the protests, and the White House has said he is open to proposals.

Not since passage of the now-controversial crime bill in 1994 has Congress tackled police conduct in significant legislation, according to congressional aides and outside experts. That bill included provisions from the late Rep. John Conyers (D., Mich.) that authorized the Justice Department to begin pattern-or-practice reviews investigating systemic misconduct within a police department.

Mr. McConnell said he planned to take his cues from Mr. Scott and the group working with him.

“None of us have had the experience of being an African American in this country and dealing with this discrimination, which persists here,” Mr. McConnell said after the lunch. “The best way for the Senate Republicans to go forward on this is to listen to one of our own, who’s had these experiences.”

Senate Minority Leader Chuck Schumer (D., N.Y.) said Tuesday that Democrats wanted to see their full policing-overhaul plan—not just pieces of it—voted on the Senate floor before the July 4 recess.

He also expressed skepticism about whether Republicans would follow through, pointing to past experiences with gun-violence legislation.

“McConnell said our caucus is going to discuss it, we are going to deal with it, etc., and then they never did. I am worried the same thing would happen here,” Mr. Schumer said.

Senate Republicans said they were open to a variety of proposals, some of which were included in a sprawling bill unveiled Monday by congressional Democrats, or are similar, but expressed caution about others. Several Republicans said they would support banning chokeholds, which would be barred at the federal level under the Democratic bill. Sen. Shelley Moore Capito (R., W.Va.) said she would like to create a national registry so that an agency hiring a new officer could check for any problems in his or her past.

Sen. Mike Braun (R., Ind.) said he would consider curbing the “qualified immunity” protection established by the Supreme Court that shields police and other government officials from legal liability for actions they take on the job.

“Most in our conference don’t want to go that far. But I’m not ruling it out. I want to see if I can get a few others interested in looking at that as well because I think that’d be the one thing that would show in our conference, we mean business,” Mr. Braun said.

Sen. Josh Hawley (R., Mo.) said he would have concerns about limiting that protection.

“I’d be really worried about wholesale changes there,” he said. “If you take away their ability to do their job, and you make them legally vulnerable for it…that makes me very worried.”

In the House, Minority Leader Kevin McCarthy (R., Calif.) and Rep. Jim Jordan of Ohio, the top Republican on the House Judiciary Committee, are leading an effort to craft legislation expected to be similar to Mr. Scott’s proposals, according to GOP aides. Their discussions are centered on strengthening police training, giving incentives for police departments to provide more use-of-force data and making sure problematic officers can’t jump to new departments, aides said.

“I do see a role for legislation,” said Rep. Adam Kinzinger (R., Ill.), who said he would like to see an increase in the use nationwide of body cameras.

Police unions signaled some openness to changes in police practices. In the past, they have opposed measures they believed would hinder officers’ ability to do their jobs or expose them to liability.

Patrick Yoes, national president of the Fraternal Order of Police, said in a statement that after an initial review of the Democratic bill, his group was “heartened to see that there were provisions in the bill that we believe, after good faith discussions, will create a law that will have a positive impact on law enforcement and policing in our country.”

“When our citizens do not feel safe in the presence of police, that’s a problem—and the FOP intends to be part of the solution,” he said.

Civil-rights advocates who have called for police-reform legislation said they often encountered strong resistance from police unions and their lobbyists on Capitol Hill.

“As we looked at policies, whether it’s racial profiling or use of force that would help us provide solutions to these awful and sometimes deadly problems, the pushback has been extraordinary,” said Hilary O. Shelton, who runs the NAACP’s government affairs office on Capitol Hill.

Write to Kristina Peterson at kristina.peterson@wsj.com and Andrew Duehren at andrew.duehren@wsj.com