Democratic Sen. Chuck Schumer of New York said last week that he was in talks with colleagues -- including several who are ranked highly by the NRA -- on possible legislation to expand background checks on private gun sales.

Sources close to both Schumer and Republican Sen. Tom Coburn of Oklahoma told CNN the two were in serious discussions about co-sponsoring a bill to strengthen background checks. Schumer sits on the Judiciary Committee, while Coburn is a former member.

However, NRA chief executive Wayne LaPierre told the panel that the current background check system doesn't work, so expanding it would only create an unmanageable government bureaucracy instead of reducing gun crime.

During the Super Bowl on Sunday night, a group called Mayors Against Illegal Guns broadcast an ad showing the NRA's LaPierre, in 1999, endorsing the expanded background checks his group now opposes.

Supporters of gun control argue that the constitutional right to bear arms can be limited, for example, by the existing ban on private citizens possessing grenade launchers and other military weaponry.

However, Denver University law professor David Kopel said last week that the Supreme Court made clear that gun control could not include weapons used commonly by law-abiding citizens, such as the top-selling AR-15 that Feinstein's legislation would ban.