Jim Kessler of the centrist-Democratic think tank Third Way, who was formerly with Americans for Gun Safety, called the records "critically important."

"It is very difficult to enforce a universal background check law without being able to ask someone to prove that they did the background check," he said.

Coburn's office did not return a request for comment, but his argument is that enforcement would come via stings on gun sellers. Many Republicans have tied record-keeping to a national gun registry, which Schumer says he opposes.

"I don't think we're that close to a deal, and there absolutely will not be recordkeeping on legitimate, law-abiding gun owners in this country," Coburn said on "Fox News Sunday" over the weekend. "And if they want to eliminate the benefits of actually trying to prevent the sales to people who are mentally ill and to criminals, all they have to do is create a recordkeeping, and that will kill this bill."

The other two senators in the group of four seeking a compromise are Democrat Joe Manchin of West Virginia, who is supported by the National Rifle Association, and Republican Mark Kirk of Illinois, who, like Schumer, has an "F" rating from the gun lobby group.

The hope is that the coalition will bring enough lawmakers with them to reach the 60 votes necessary to pass a compromise bill. The four have already been reaching out to other senators for input on the legislation, including Sen. John McCain, R-Ariz., who said this week he is in discussions with the core group. Manchin, meanwhile, presented a rosier image of negotiations than Coburn, saying, "We're working. We'll get there, we'll get there."

The American people widely support universal background checks, with numerous polls showing that nine in ten back the idea, including the vast majority of NRA members. At the moment, gun dealers are required to perform background checks but not sellers at gun sales or personal sellers. People can fail the check if they have a restraining order against them or have been convicted of a violent crime, among other reasons.

The NRA released an ad Monday accusing Schumer of favoring a gun registry, despite his claims to the contrary. It spotlighted Schumer's comment saying he had been pushing "universal registration" to make the claim. Schumer's spokesman told CBS News that the comment was a misstatement and noted that he had used the correct phrase -- universal background checks - elsewhere in the interview. The ad appeared to be an attempt to gin up opposition to a universal background check bill by casting it as a backdoor attempt to create a registry.

Kessler criticized the NRA for the suggestion that Schumer is pursing a national gun registry. He pointed out that records for 170 million guns that have gone through a criminal background check are already sitting in gun stores, and noted that people do not consider those records akin to a federal registry.