NEWTOWN — Victor Benson will never forget the man who walked into his shop and said, “I want a gun to shoot my son-in-law.”

Benson didn’t care whether the man was joking or not.

“You just don’t walk in and say something like that,” said Benson, owner of the Freedom Shoppe in New Milford for the last 17 years. “I memorized what he looked like, and he’s never buying a gun from me.”

Such encounters are rare for Benson, who knows 90 percent of his customers. But when strangers do walk in, Benson is like any other gun store owner who needs to know that his buyers are mentally fit and law-abiding, and he said anything that closes gaps in the federal background check process for gun shop purchases gets his vote.

The last thing any merchant wants is to approve a sale to someone who shouldn’t have a gun because that person’s name was never entered in the federal database. That’s what happened in Texas on Nov. 5, when a convicted domestic abuser whose records were never submitted to the background database was able to buy guns and massacre 26 people in a church.

“I don’t want to be that guy who sells a gun to some nut,” Benson said.

Support for better background checks from Benson and thousands of merchants like him in Connecticut and across the country is one reason for all the discussion last week about bipartisan bills to close gaps in the National Instant Criminal Background Check System.

Another reason: It’s so rare for Democrats and the firearms industry to find common ground that when they do, it almost seems too good to be true.

“I think this is clearly a sweet spot for an agreement,” said Sen. Chris Murphy, a Democrat who is co-sponsoring one of the bills with Texas GOP Sen. John Cornyn. “The politics on this are changing.”

The legislation has even made temporary allies out of Sandy Hook Promise, the Newtown-based anti-gun violence nonprofit started by families who lost loved ones in the 2012 Sandy Hook massacre, and the National Shooting Sports Foundation, the Newtown-based trade association for the firearms industry.

Each group praised the Senate bill’s approach mixing support and enforcement to encourage agencies to fully report criminal and mental health records.

The “Fix NICS” bill would withhold funds from states and federal agencies that don’t fully report records of people who are barred from owning guns, and calls for priority in federal grantmaking for those that do comply. The bill would also provide money and technical assistance to help standardize varying definitions that now make reporting cumbersome.

The aim is to prevent gun sales to thousands of people with convictions for violent crimes and unstable mental health histories whose information has not been entered into NICS.

That work has already been started by NSSF, which has successfully lobbied 16 states to pass laws requiring better reporting practices. As a result, 2.8 million mental health records have been added to NICS in the last three years, said Jake McGuigan, director of governmental relations and state affairs at NSSF.

“In 2012, Massachusetts had one record entered into NICS,” McGuigan said. “Today Massachusetts has more than 14,000 records entered in the system.”

The House bill vote might sound like something all sides could welcome, but Democratic support was eroding on Friday because of fears that it would be packaged with another GOP bill making it easier for people with permits to carry concealed guns across state lines — a measure opposed by Democrats and gun violence prevention advocates in Newtown.