A group of House Republicans is calling on House Speaker Paul Ryan Paul Davis RyanKenosha will be a good bellwether in 2020 At indoor rally, Pence says election runs through Wisconsin Juan Williams: Breaking down the debates MORE (R-Wis.) to bring to a vote a measure aimed at bolstering the accuracy of the nation's background check system for gun buyers.

In a letter led by Rep. Leonard Lance Leonard LanceThomas Kean wins GOP primary to take on Rep. Tom Malinowski Gun debate to shape 2020 races GOP fears Trump backlash in suburbs MORE (N.J.), the GOP lawmakers called on Ryan to allow "immediate consideration" of the Fix NICS Act, which would push federal and state agencies to comply with existing laws and more accurately report criminal records to the National Instant Criminal Background Check System (NICS).

"The Fix NICS Act, would greatly improve the sharing of mental health and criminal record information between state and local agencies and the federal background check database," the lawmakers, many of whom are members of the moderate Tuesday Group, wrote.

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"It has broad bipartisan support in both the House and Senate. Let us pass this legislation and take an important step toward making our Country safer from gun violence."

White House press secretary Sarah Huckabee Sanders says that President Trump Donald John TrumpUS reimposes UN sanctions on Iran amid increasing tensions Jeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Trump supporters chant 'Fill that seat' at North Carolina rally MORE supports the measure, though she emphasized that "discussions are ongoing" earlier this week.

The bill was first proposed last year after it was revealed that the Air Force had failed to report to the FBI the criminal history of a man who opened fire on a church in Sutherland Springs, Texas, killing 26 people.

That shooter, a former member of the Air Force who faced court-martial over bad conduct, would have been barred from purchasing guns had the information been properly reported.

The House passed the Fix NICS Act last year, but remains pending in the Senate Judiciary Committee. The renewed call for its passage comes days after a deadly shooting at Marjory Stoneman Douglas High School in Parkland, Fla., near Fort Lauderdale. That attack left 17 people dead and 14 others injured.

The Parkland shooting reignited a ferocious national debate over gun control reform and sparked widespread calls to ban assault-style weapons and strengthen the nation's background check system for gun buyers.

Democrats consider Lance vulnerable in the 2018 midterm elections. Former presidential nominee Hillary Clinton Hillary Diane Rodham ClintonJeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Momentum growing among Republicans for Supreme Court vote before Election Day Warning signs flash for Lindsey Graham in South Carolina MORE won his district in 2016, and Democrats have been looking for candidates to challenge him. The Cook Political Report currently rates Lance's district as "lean Republican."

Trump signaled support this week for strengthening background checks and raising the minimum age for purchasing firearms. But he has also voiced support for arming trained teachers and doing away with gun-free zones around schools.

--Updated at 5:03 p.m.