(CNN)The second-ranking Senate Republican broke with President Donald Trump on Friday when he told CNN that the President's proposal to raise the legal age to buy a rifle to 21 may not "save lives" and doesn't get at the "root of the problem."
No. 2 GOP senator: Raising age limit for rifles may not save lives, nor have enough votes to pass
Sen. John Cornyn of Texas also said that the idea, which Trump unexpectedly embraced this week, may not have enough GOP support to pass the Senate.
Cornyn said Washington -- which is searching for a legislative response to the massacre last week at a high school in Florida -- should instead focus on passing his more narrow bill, known as "Fix NICS," that would shore up reporting compliance to the National Instant Criminal Background Check System.
Cornyn oversees counting and whipping votes for the GOP leadership. He said he thought other Republicans would share his concerns making it uncertain if Trump's proposal could get the 60 votes it would need to pass the Senate.
"I have not talked to my colleagues about it, but I imagine some of the concerns I've raised would be shared by others," he said. "My attitude is let's get that (Fix NICS) deal done and then if there are other things we need to work on after we've gotten that signed into law, let's keep working."
To date, three GOP senators -- Marco Rubio of Florida, Pat Roberts of Kansas, and Jeff Flake of Arizona -- have expressed support for the age increase.
"I think what we want to focus on is things that will actually save lives," Cornyn said when asked if he supported raising the age limit. "That's why I think the focus should be on the Fix NICS bill, which is the only bipartisan piece of legislation that can be signed into law."
"There are a lot of other ideas out there that people are proposing and that I don't think will actually change any outcomes," he added.
Congress is in recess until next week. Cornyn, who was in the Capitol on Friday to preside over a brief pro forma session, said he thought an age restriction would be problematic if an 18-year-old Marine or police officer were told he or she could not buy a gun.
"I can see that it would be difficult to enforce," he said. "I'm not sure why we would go to those lengths when I don't think that gets to the root of the problem."
Republicans hold a 51-49 advantage. It's likely many Democrats would support raising the limit, but it's a relatively new discussion point, so it's not clear where they all stand.
Cornyn was asked if Trump is getting ahead of most Hill Republicans on gun control and whether he is concerned Trump is driven by an emotional response to the mass killing and the dramatic listening session he held with survivors this week at the White House.
"My perception is he's brainstorming and throwing out possible ideas. There's nothing wrong with that," Cornyn said. "But, obviously, that time at the White House was very emotional with the victims and the families, and I thought it played it to his strength."
"I don't know that many other people who could pull that off, including other past presidents," Cornyn added.
Finally, Cornyn said there are discussions underway between Democratic and Republican leadership to try to advance the Fix NICS bill by unanimous consent, meaning without a roll call vote, as early as Monday. But he said as of now, there is not an agreement to do that.
"I will be fine with that if we can get it, but I don't think we're there yet," he said.
The House passed its own version of Fix NICS but tied it to a controversial conceal carry measure that Democrats strongly oppose and would object to if the Senate tried to take it up.