The measure is reminiscent of another law, passed in 2007 after a massacre at Virginia Tech, which provided incentives and grant money for states to update the background-check database with more criminal records and mental health information. That deal was a delicate compromise between gun control advocates and the National Rifle Association, but as the Sutherland Springs shooting showed, it fell short of its promise.

The new measure, called the Fix NICS Act — NICS is the acronym for the National Instant Criminal Background Check System — is modest when compared to the demands of gun safety advocates, who have for years been pushing to expand background checks for gun buyers and to limit the purchase of assault weapons.

Instead, the new measure would require federal agencies and states to produce plans showing how they intend to comply with existing laws governing reporting to the background check system. It would reward states that comply with financial incentives by making it easier for them to get federal grants. And it would penalize federal agencies that fail to report to the system by barring bonus pay for political appointees.

John Feinblatt, the president of Everytown for Gun Safety, an advocacy group here, praised the measure as “an important first step” to improve the background check system.

The massacre in Texas showed once again that the computerized background check program is only as good as the data that is put into it. The 2007 massacre at Virginia Tech brought attention to the system’s flaws after it was revealed that the gunman, Seung-Hui Cho, had been ordered by a court into mental health treatment, but that court order had not been forwarded to the background check database, a failure that may have allowed him to purchase his arsenal.

Although the gun debate is contentious in Congress, surveys show the majority of Americans — even those in households with guns — want to see gun safety laws passed. A poll released Thursday by Quinnipiac University found that 60 percent of voters favor stricter gun laws, and 59 percent say it is too easy to buy a gun in the United States. The survey also found that 94 percent of those living in households with guns favor universal background checks for gun buyers.