(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."

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."

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