Sen. John Cornyn (R-TX) said Thursday he wants to have an “informed debate about reforms we can make” to federal gun policy in the wake of a mass shooting at an elementary school in his state.

Senate Democrats are eager to vote on a gun bill in response to the shooting but face steep odds to do so because any effort to pass new restrictions or reforms to existing laws would have to win bipartisan support in the Senate in order to reach the upper chamber’s 60-vote filibuster threshold and go to President Joe Biden’s desk.

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Senate Majority Leader Chuck Schumer praised Sen. Chris Murphy (D-CT) on Thursday for reaching out to Republican colleagues to see what types of legislation they would support but expressed skepticism that his bipartisan talks would succeed and indicated he intends to vote on the matter on the chamber’s return from its Memorial Day recess with or without its support.

But some Senate Republicans have indicated they are open to debate about certain types of reforms, including red flag laws and enhanced background checks.

In remarks on the Senate floor, Cornyn, who returned to the Senate after visiting Uvalde, signaled his openness to such a debate. Cornyn said he donated blood and received a briefing from local authorities on the shooting at Robb Elementary School that left at least 19 children and two adults dead. He opened his comments by noting Thursday was supposed to be the last day of school in Uvalde.

“I can't fathom the pain of seeing an empty bed where your child slept the night before,” he said. “The obvious question is, how do we prevent something like this from happening again?”

Cornyn said he wants to allow law enforcement the time they need to conduct a thorough investigation of what took place so the Senate can have a debate informed by the facts of the case.

“I'm eager to see whether there were any gaps that might have done something to make this attack less likely, that might have even prevented this attack from taking place,” he said.

Cornyn said he has worked with colleagues in the past on measures to “prevent senseless tragedies from occurring,” pointing to the Fix NICS Act and the Mental Health and Safe Communities Act.

“I'm not interested in making a political statement,” Cornyn said. “I'm not interested in the same old tired talking points. I'm actually interested in what we can do to make the terrible events that occurred in Uvalde less likely in the future.”

Once all the facts of the shooting are established by the investigation, Cornyn said, he wants to have an “informed debate about reforms we can make.”

“I look forward to participating in those discussions,” Cornyn said.

Cornyn was among Republican lawmakers scheduled to speak at an NRA convention this weekend in Houston but will reportedly no longer do so, citing a scheduling conflict. Sen. Ted Cruz (R-TX) told local media this week he still plans to attend.

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Cornyn also praised the community of Uvalde for the response to the shooting, saying he met a man who came to donate blood who lost two of his grandchildren in the attack.

“It’s not possible for us to comprehend the pain he must be feeling, but there he was, donating blood,” Cornyn said. “Two young lives were stolen from his family, but still, this man stood in line so he could support his community in some tangible way.”