WASHINGTON – At a bipartisan White House meeting on gun violence Wednesday, President Donald Trump pressed Republican U.S. Sen. John Cornyn of Texas and other lawmakers to come together and create “one great piece of legislation” to address background checks on gun sales, in addition to other measures.

The meeting was held in the wake of another national reckoning on gun violence following the mass shooting of a South Florida high school on Feb. 14. Seated next to the president, Cornyn said it was “unacceptable” for lawmakers to leave Washington “empty-handed.”

“The public demands that we act,” Cornyn said. “In the past, we’ve acquiesced to failure and have not done things we know were within our power to accomplish.”

Cornyn, alongside U.S. Sen. Chris Murphy, a Connecticut Democrat, has been pushing for a vote on his “Fix NICS” bill, which would hold government agencies accountable if they fail to upload individuals’ criminal histories to the FBI’s National Instant Criminal Background Check System. Forty-six senators have signed on to co-sponsor the bill, including 10 this week.

Trump praised the bill, but added it “would be nice if we could add everything on to it,” referencing raising the age limit on purchasing an AR-15 rifle from 18 to 21, among other proposals.

“Your bill is really good and really important, having to do with a certain aspect, but maybe we could make it much more comprehensive and have one bill instead of 15 different bills,” Trump told Cornyn.

Cornyn replied: “If we can get 60 votes on it, Mr. President, I’m all for it.”

Congressional leaders have clashed over the "Fix NICS" proposal, with Senate Democrats arguing the legislation does not go far enough in addressing gun violence or problems with the background check system.

“Rather than just passing one narrow bill and moving on, we Democrats intend to push our Republican colleagues to have a real debate on gun safety,” Senate Minority Leader Charles Schumer of New York told reporters Tuesday, according to Politico.

Trump also repeatedly directed Cornyn and Murphy to work with two other senators prominent in the gun control debate: Republican Pat Toomey of Pennsylvania and Democrat Joe Manchin of West Virginia. The pair authored a 2013 bipartisan bill to expand background checks on gun sales. That measure failed in the Senate with 54 votes, but there have been calls for it to be revived following the shooting in Parkland, Florida.

“If the four of you could work together and come up with some beautiful foundation, add and subtract to it, put it up for a vote, let’s get it done,” Trump said toward the end of the meeting. “That’s what we have to do.”

Senate Republican leadership has opposed marrying the Toomey-Manchin legislation with the Fix NICS bill, arguing that a narrower measure with broad bipartisan support is crucial to passing any legislation that addresses gun violence.

Cornyn repeated that point Wednesday, telling Trump and other legislators gathered around the table that Congress should act to address what it can now.

“We need to get started on things that only we can do, which would be this background check system,” Cornyn said. “I would just like to recommend to my colleagues here: Get that done and we build on it. We don’t stop there, we build on it.”