Sen. Tim Scott addresses a crowd at a town hall meeting at the Charleston County Council Chambers in February 2017 in North Charleston, South Carolina.
CNN  — 

Republican senators – confronted by marches in cities and small towns across America – are growing more vocal that the party must act on policing reform, a stark contrast to just a week ago when many members dismissed the idea of the Senate acting at all.

On Tuesday, Sen. Tim Scott of South Carolina, the only black Republican in the US Senate, presented his ideas for policing reform to colleagues during a closed-door lunch. The proposals ranged from a nationalized database chronicling police misuse of force incidents as well as broader incentives to help local and state police forces implement bias and intervention trainings.

“Reforms seem to be in order. You asked if this is a moment and I’d say it is,” Sen. Kevin Cramer, a Republican from North Dakota, told CNN. “A crisis creates an image and an opportunity that is ripe for discussion. It seems like now is exactly the time to have a discussion about it.”

White House chief of staff Mark Meadows and President Donald Trump’s son-in-law Jared Kushner also met with Scott on Tuesday afternoon. Meadows took said it was a “good conversation” and it’s a work in progress.

There is now a five-person task force that will lead Republicans in their effort to craft legislation including Scott, Senate Judiciary Chairman Lindsey Graham, also of South Carolina, John Cornyn of Texas, Ben Sasse of Nebraska and Shelley Moore Capito of West Virginia. All of the members are working just days after Democrats unveiled their own proposal, which went further in dictated specific changes that state and local police forces should make including banning chokeholds.

The effort by Senate Republicans to debate police reform marks a swift turnaround from not long ago when many GOP lawmakers in the upper chamber were dismissive about such a possibility.

Attitudes appear to have shifted, however, amid the sustained and intense pressure felt by lawmakers as many Americans have taken to the streets in protest over police misconduct and racial injustice after George Floyd was killed while in Minneapolis police custody.

Senate Majority Leader Mitch McConnell said on Tuesday, “What we’ve been talking about here in the Senate Republican conference is what we think is the appropriate response to the events of the last few weeks, and under the leadership and guidance of Sen. Tim Scott, at some point in the near future we’ll have recommendations to be made.”

Any effort to craft a police reform proposal that can win widespread support from Senate Republicans is likely to prove a politically thorny task and there is no guarantee that such a proposal would win bipartisan support as Democrats push their own legislative reform package.

“It’s my view that the best reforms need to happen at the local level. That is where the community can help drive them as opposed to just one national standard across the board. If there is something we can do here that makes sense, I will certainly look at it,” Sen. Marco Rubio, a Republican from Florida told CNN, adding, “I also think we make a big mistake if we think this entire issue is just a police issue. This is about a whole lot more than just police issues.”

Sen. John Thune, a South Dakota Republican and the majority whip in the chamber, also pointed to local officials to change policing as opposed to a federal leeal solution.”

“The federal government doesn’t run a police department. The local mayor, city council, and police chiefs do,” Thune said in response to a question about outlawing chokeholds.

He added, “To my knowledge, most of the police department have a ban on some of those practices like chokeholds already, but I’m for looking at all that. But I want to look more broadly, not just at the tactics.”

During a Republican Senate lunch on Tuesday, Scott did not lay out a definitive proposal, but instead helped facilitate a conversation on what proposals members could be open to. The current proposal does not include a ban on chokeholds, but Capito said they are still hearing suggestions. Cornyn said that one of the options would be creating a policing commission to study best practices in policing.

Cornyn also said there is broad discussion about creating a national database that tracks misuse of force incidents.

Scott discussed the potential legislative proposal on police reform with reporters Tuesday following his presentation to members during the Republican policy lunch. He said it would include collecting data from all police departments, ensuring the use of body cams, various trainings like de-escalation training and bias training, limiting use of force that leads to death or serious bodily injury, and the Senate’s anti-lynching bill, among other components.

Asked about a timeline for the bill, Scott said he’s an “optimist” and thinks they can get something done very soon because “time is of the essence.”

He noted, however, that McConnell “will decide what to do with the legislation once we submit it.”

Scott continued, “I think we have an opportunity to see something happen because I think time is of the essence. I think 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, we’re responding in a positive, constructive manner.”

House Republicans, meanwhile, are planning to introduce their own policing plan, led by Rep. Jim Jordan of Ohio, a source familiar tells CNN. Jordan, the top Republican on the Judiciary Committee, is working with Minority Leader Kevin McCarthy’s office, GOP Whip Steve Scalise, and other members in the conference to write the legislation.

The source said the bill is “in the very early stages” and members are “considering all options.”

Senate Democratic leader Chuck Schumer did not say on Tuesday that Senate Democrats would get behind a Republican reform proposal – and signaled skepticism about the effort.

“We don’t know what it is yet,” Schumer said when asked about the Senate GOP effort. He added, “I am really worried. Remember when there was a lot of gun violence and McConnell said, oh our caucus is going to discuss it, we’re going to deal with it, etc, and then they never did. I’m worried the same thing will happen here.”

CNN’s Haley Byrd, Manu Raju and Jim Acosta contributed to this report.