Senate Majority Leader Mitch McConnell (R-Ky.) on Tuesday called on the Senate to try to make progress on “bills we agree on” following the Florida high school shooting that reopened the debate on gun control.

The GOP leader said there is broad support in his conference for background check legislation, and suggested it would seek to move forward on that issue despite calls from Democrats for more dramatic steps.

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McConnell signaled his support for a bipartisan bill sponsored by Senate Republican Whip John Cornyn (Texas) and Democratic Sen. Chris Murphy (Conn.) to give states more incentive to submit information to the National Instant Criminal Background Check System.

McConnell says there is broad support in his conference that “we should try to make some progress on the bills we agree on.”

He acknowledged that the Cornyn-Murphy legislation is not a “panacea” but instead a good, bipartisan start.

McConnell also signaled support for making schools more secure, even though he pointed out that it’s more of a local than federal issue.

“We ought to be able to harden those schools and protect our youngsters,” he said.

He noted that people are required to pass through metal detectors to enter the Capitol and to board airplanes.

The House approved background check legislation last year, but coupled it with a measure that allows concealed carry permits to be valid across state lines. The combined bill has not passed the Senate because of opposition among Democrats to the concealed carry language.