President Trump is open to improving the gun background check system, according to a statement released by the White House Monday morning.

"The President spoke to Senator Cornyn on Friday about the bi-partisan bill he and Sen. Murphy introduced to improve Federal Compliance with Criminal Background check Legislation," White House press secretary Sarah Sanders said. "While discussions are ongoing and revisions are being considered, the President is supportive of efforts to improve the Federal background check system."

A spokesman for Cornyn told CBS News' Alan He in a statement, "It's clear the president is interested in getting something done," and he pointed out that the bill has passed the House, is endorsed by the NRA and is cosponsored by both leaders in the Senate, Sens. Mitch McConnell, R-Kentucky, and Chuck Schumer, D-New York.

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In November, Cornyn, with a bipartisan group of senators, introduced the "Fix NICS Act" to ensure federal and state authorities accurately report relevant criminal history records to the National Instant Criminal Background Check System (NICS).

The bill would penalize federal agencies that fail to properly report relevant records and incentivize states to improve their overall reporting. It would also direct more federal funding to the accurate reporting of domestic violence records. That bill was introduced after a mass shooting in a small church in Texas left 26 people dead.

Last week, 19-year-old Nikolas Cruz allegedly shot and killed 17 people at a school in Parkland, Florida with a gun he had purchased last February. The president visited the wounded and families of the victims Friday.