"I like taking the guns early", he said, adding, "Take the guns first, go through due process second". "His approach to this is that measure along with fixing the background check system, strengthening it", Shah said.
Some lawmakers flatly rejected the concept, which Trump previously said should be left up to states and school boards.
"Let's do what we can and build from there", Cornyn said.
Sen. Roy Blunt, another member of Republican leadership, said he also expects an open amendment process to go forward.
The 14 students and three adult educators killed were shot with a semi-automatic AR-15-style assault weapon, which the authorities say was purchased legally previous year by the accused gunman, Nikolas Cruz, when he was 18.
"But you have to be very, very powerful on background checks", he continued. Pat Toomey of Pennsylvania, arguing that raising the age limit - something the National Rifle Association opposes - was "something we have to think about".
"Despite the aggressive opposition of the NRA, Senator Toomey never wavered in his commitment to this legislation", Toomey press secretary Steve Kelly said in a statement. One bill to do that passed the House, but it has not moved in the Senate because it also would expand Americans' ability to carry concealed weapons.
Some House Republicans are resistant to separating the two issues, as is likely in the Senate.
Trump's session with the governors will be the latest in which he solicits ideas for stopping gun violence at schools as the White House works to finalize an expected legislative proposal. And you know what?
Democrats and some senior Republicans in the Senate are expecting votes on a variety of gun control proposals in the coming days. "People aren't bringing it up because they're afraid to bring it up". Trump says, "We have to do something about it".
"I don't want teachers to have guns".
In 2013, the NRA said the Toomey-Manchin bill would "not prevent the next shooting" and would not "solve violent crime".
Trump has floated numerous ideas since the shooting, including raising the minimum age for the purchase of assault-style weapons, improving background checks for gun purchases, arming educators and paying them bonuses, and re-opening mental institutions. But Trump, Democrats and some other Republicans, are now pushing for broader measures to be added, which could stall progress. And they're pressing him to endorse legislation that would extend the pre-purchase checks at least to online and gun show sales.
During the meeting Trump made other policy suggestions, including making it harder for mentally ill people to buy guns.
"It can be ended and it will be ended", Trump said.
Trump approved of many suggestions from Democratic lawmakers.