However, an identical measure failed earlier this year in the Senate when it didn't receive enough Republican votes. It was unclear whether Thompson's legislation, co-authored by Rep. Pete King, R-N.Y., would fare better.
"It's going to be hard," said Thompson, chairman of the House Gun Violence Prevention Task Force. "I'd like to get it before the next election."
In an effort to show broad voter support, Thompson hosted Kelly and local gun owners at the firing range Friday.
They all shot clay pigeons. Thompson, a Vietnam veteran, proved to be a dead shot in pink-tinted safety glasses. Kelly wasn't half-bad, either.
"I figured you could shoot," Thompson told him.
Others voiced their support for the two men.
Matt Schiefferly, who owns a Napa gun shop, said background checks already are required in California, but it is too easy for people to drive across state lines or get on the Internet and buy guns.
Thompson's bill would change all that, he said.
"It's something I think we need," he said.
Susan Rinehart, a gun club member in Napa, Sonoma and Mendocino counties, said everyone she's talked to is for it.
"I think it's a wonderful idea," the retired government worker said.
Meanwhile, Kelly and Giffords have been spreading the word in other parts of the country. He said the couple just completed a weeklong trip, hitting seven states.
His wife, who has difficulty speaking and walking, and who has lost the use of her right arm, still shoots at targets occasionally, albeit it with her uninjured left hand, he said.
"If you got in a big car wreck, are you going to stop driving a car?" he said.
You can reach Staff Writer Paul Payne at 568-5312 or firstname.lastname@example.org.