Senate Republicans are signaling an openness to talks with Democrats on gun violence in the aftermath of back-to-back mass shootings.

As pressure builds around gun reform on Capitol Hill following the shootings, Republican leaders have encouraged talks exploring a potential bipartisan course of action amid discussion on measures ranging from “red flag” legislation to background checks.

Senate Minority Leader Mitch McConnell (R-Ky.) told CNN on Thursday that he tapped Sen. John Cornyn (R-Texas), who recently returned to his home state after the recent shooting at an elementary school in Uvalde, to participate in bipartisan talks.

“What I’ve asked Senator Cornyn to do is to meet with the Democrats who are interested in getting a bipartisan solution and come up with a proposal, if possible, that’s crafted to meet this particular problem,” he said, stressing proposals directly related to the Uvalde shooting.

Other GOP leaders have also said they’re supportive of bipartisan talks, while similarly pushing back on broad proposals that stray in focus from recent gun violence.

“I think that we have to at least listen to each other and see if there’s a path forward where we might be able to find solutions that actually address the problem,” said Sen. John Thune (S.D.), the No. 2 Republican in the upper chamber.

Many Republicans were resistant to Democratic-led calls for gun reform last week, following a racist shooting at a store in a predominantly Black neighborhood in Buffalo, N.Y., that killed 10 people.  

But Democrats say more Republicans are expressing interest in a possible compromise as the issue of gun violence garners more public attention in the days following the Texas shooting, which left 19 children and two teachers dead

Sen. Richard Blumenthal (D-Conn.), who has been helping lead bipartisan talks around gun reform along with Sen. Chris Murphy (D-Conn.), estimated on Thursday that about six to twelve Republicans have “indicated serious interest,” particularly on red flag 

legislation.

According to Everytown for Gun Safety, a national group focused on preventing gun violence, nineteen states have implemented such laws designed to keep people at risk of harming themselves or others from temporarily accessing firearms.

There’s also interest in pursuing proposals to expand background checks, like the bipartisan legislation Sens. Joe Manchin (D-W.Va.) and Pat Toomey (R-Pa.) previously brought up years back that sought to mandate background checks for all gun commercial sales.

“That is the centerpiece as far as I’m concerned of what might be done, and frankly, there’s not much more than that – that can be done,” Toomey told The Hill on Wednesday, adding that a red flag is “possible,” but would be a “tough” lift.