The White House has said that US President Donald Trump is "supportive" of efforts to improve background checks on gun ownership.
A statement said he has spoken to Republican Senator John Cornyn about a bipartisan bill he helped introduce.
The 2017 bill sought to improve federal compliance with checks that are processed before someone can buy a gun.
It comes after authorities said the suspect in last week's school shooting in Florida had bought his gun legally.
The National Instant Criminal Background Check System (NICS) currently relies on state and federal officials to report any criminal convictions and mental health issues that should legally stop someone buying a gun.
Its failings were put in the spotlight last year after the US Air Force admitted it had failed to flag a gunman's domestic violence conviction before he shot dead 26 people at a church in Sutherland Springs, Texas.