Washington(CNN) Senate Minority Leader Mitch McConnell told CNN on Thursday he met earlier in the day with Texas Sen. John Cornyn and encouraged the senior Republican senator to begin discussions with Democrats, including Sens. Chris Murphy of Connecticut and Kyrsten Sinema of Arizona, to see if they can find a middle ground on legislation to respond to the tragic Texas elementary school shooting.
Lawmakers on both sides of the aisle are facing enormous pressure to take action in the wake of the horrific shooting, but members on both sides acknowledge the uphill battle to find common ground given the highly polarized political climate around gun legislation and widespread GOP opposition to stricter gun control.
It is significant, though, that McConnell has decided to weigh in and is giving a greenlight to a bipartisan effort on a potential legislative response to the shooting. But it still remains to be seen what, if anything, talks will amount to given that countless mass shootings in recent years have failed to break the partisan stalemate over the issue of gun policy in Congress.
McConnell would not say specifically what the contours of that legislation should be, instead signaling he wants Cornyn to be the one to negotiate.
"I met with Senator Cornyn this morning. As you know he went home yesterday to see the family members and begin the fact finding of this awful massacre and I have encouraged him to talk with Sen. Murphy and Sen. Sinema and others who are interested in trying to get an outcome that is directly related to the problem. I am hopeful that we could come up with a bipartisan solution," McConnell told CNN.
Legislation to respond to mass shootings and counter gun violence has long been one of the most divisive and intractable policy issues facing lawmakers.
Cornyn said earlier on Thursday that he is going to meet today with Murphy, one of the most prominent Democrats on the issue of gun policy, to see what can be worked on. He and Murphy have actually met on and off for the last year on this issue and are going to try and resume some of their talks.
He said he doesn't know if they can get there on background checks or other bills, but he did say "this hopefully will provide a new, greater sense of urgency."
Cornyn has also said that he does not want to see the shooting used as a way to infringe on gun rights.
"There's a whole list of things that we can consider, but I think particularly mental health, access to mental health treatment is high on that list," he said, adding, "I think we need to be open to whatever, wherever the evidence leads us. I would say that this is not an excuse to infringe the Second Amendment rights of law abiding citizens doing that will do nothing to fix tragedies like this."
Later on Thursday Cornyn said he had "touched gloves 'with Murphy and that their talks are "just getting started."
Senate Majority Leader Chuck Schumer has indicated that Democrats are willing to give some time and space for efforts to reach some kind of bipartisan compromise on gun legislation, though he has noted the odds are long. He has also made clear that these efforts will not be given an unlimited amount of time to play out.
"We have to try everything. We must not leave a single stone unturned," Schumer said earlier on Thursday.
He added, however, "this is not an invite to negotiate indefinitely. Make no mistake about it, if these negotiations do not bear fruit in a short period of time, the Senate will vote on gun safety legislation."
Senate Democrats took steps on Tuesday night to place two House-passed background checks bills onto the legislative calendar so they can be voted on. It's unclear, however, when the Senate might vote on the measures and they would not be expected to overcome a GOP filibuster.
Murphy described the effort he is engaged in, saying, "we'll have a series of meetings today to set the table for next week and then we will work really hard to try to find a bipartisan product."
He sounded a note of caution similar to Schumer however, adding, "I'm not going to negotiate forever."
"I've been through this before," he said. "I know sometimes Republicans are more interested in talking than doing, so if we can't get some progress by the end of next week, then I'll say to Sen. Schumer it's time to take votes. And Sen. Schumer has committed that we're gonna have votes one way or the other on the floor of the Senate on issues like background checks."
A number of Democrats are expressing optimism about the prospects, despite the extremely long odds of any kind of agreement.
Sen. Joe Manchin, a moderate West Virginia Democrat, called the bipartisan meeting on gun reform "very encouraging."
He said that members involved were looking at "a little bit of everything" -- such as background check expansion and red flag laws -- to see what could get bipartisan support.
Asked if he believed Democrats could get 10 Republicans on board, he said that he "always" believes they will. "You quit thinking about the politics and can you get elected on these votes. This is about basically protecting children. If they can't rise to that, they ought to deep, deep dig inside and find out why in the heck we're here."
Most legislation requires at least 10 Republican votes to overcome a filibuster in the Senate given the current 50-50 partisan split.
Democratic Sen. Richard Blumenthal of Connecticut, who has been engaged in gun reform talks with colleagues on both sides of the aisle for years, said Thursday when asked about red flag laws and other gun-related measures that "for the first time," he and Cornyn and other Republican colleagues are "talking seriously," after the shooting in Texas.
"That's a major cause of encouragement to me," he said. "I'm hoping now we can reach that critical mass that will get us to 60 votes. I'm hopeful for the first time in a long time."