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“What we were going to do is put at risk gun owners who actually followed the law and 11 years later can’t find a piece of paper that said they did it right and the presumption is, you’re guilty,” he said. “And the vast majority of gun owners in this country and law-abiding citizens do not want to sell a gun to somebody that shouldn’t have it. And they’ll follow this.”

The base bill will include universal background check legislation proposed by Sen. Charles E. Schumer, New York Democrat, which requires checks on virtually all gun transactions, with limited exemptions, such as gifts exchanged between family members.

Currently, all sales by licensed firearms dealers must go through background checks, but transactions between private individuals do not. Lawmakers are looking for a way to extend checks to almost all transactions without also creating a record-keeping system that gun-rights supporters fear could turn into a gun registry.

Mr. Coburn said he doesn’t believe the federal government will confiscate peoples’ guns — another fear of many gun-rights activists — but that its track record on other issues gives people justifiable pause.

“Remember, there are a lot of people in this country that — and rightly so, given the behavior of the federal government, in terms of its fiscal capability, in terms of regulatory overreach, in terms of poking its nose into every area of everybody’s life, in terms of domestic drones, in terms of all this other stuff — that you’ve created a certain level of paranoia in this country, and some of it’s justified,” he said.