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).
When asked if Mr Trump was open to reinstating a ban on assault-type weapons, White House spokeswoman Sarah Huckabee Sanders said officials "haven't closed the door on any front". "And we must to do more to protect our children".
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Trump spent the weekend in South Florida, only an hour's drive from Stoneman Douglas High School where 17 people were fatally shot last week.
Past year the President told a National Rifle Association convention that he would "never, ever infringe" on the constitutional right to keep and bear arms. "We must actually make a difference". "No special interest group, no political agenda is more critical than timely passage of legislation to effectively address the gun violence issues that are rampant in our country".
"Every day when I say "bye" to my parents, I do acknowledge the fact that I could never see my parents again".
The legislation does not add new restrictions on gun purchases.