"We're not proposing gun registration; we're proposing background checks for criminals," Obama declared Wednesday.
While Reid said he wanted a vote on the gun law package next week when the Senate returns from its spring break, two Democratic sources said that was unlikely because of the hangup over background checks.
Erickson Hatalsky of Third Way said negotiations under way between Republican Sen. Tom Coburn of Oklahoma and Democratic Sen. Chuck Schumer of New York would determine if a compromise on the background check provision was possible.
If so, the backing of Coburn, who gets an "A" rating from the NRA for his record on gun rights, would cause enough other Senate Republicans to join him in voting with Democrats to overcome a filibuster, she said. A compromise backed by Senate Republicans also would have more traction in the GOP-led House.
Straw purchases provision challenged
The NRA and Republicans also are challenging another provision of the Senate legislation intended to crack down on straw purchases, arguing the current language is too broad and could penalize the original seller of a weapon that passes through several hands.
Proposed by Senate Judiciary Committee Chairman Pat Leahy, D-Vermont, the measure was the one in the bill that had been given the best chance of winning eventual congressional approval.
The NRA's influence was on display Tuesday when former GOP Rep. Asa Hutchinson indicated to CNN that he differed with the NRA, then later clarified that he joined the organization in opposing the kind of expansion proposed by the Senate legislation.
"I'm open to expanding background checks," Hutchinson said initially in an interview about a panel he headed that was set up by the NRA to examine school safety following the Newtown attack.
Such expanded checks must be done "in a way that does not infringe upon an individual and make it hard for an individual to transfer to a friend or a neighbor or somebody," he added.
After the remarks, an NRA spokesman told CNN that Hutchinson was "not speaking" for the group.
Hutchinson later affirmed that he was not speaking for the NRA, and put his remarks in line with the group's position of including only more information on people with mental illness in the existing National Instant Check System.
"I have been focused on school safety and the interview surprised me by almost exclusively asking about the ongoing gun control debate," Hutchinson said in an e-mail to CNN. "On background checks, my recollection is that I noted there is insufficient data in the NICS and that needs to be fixed and expanded. I am certainly 'open' to legislation that addresses expansion of data in the NICS."
Hutchinson's task force called Tuesday for training and arming adults in schools to reduce the response time in the event of an attack like the one in Newtown.
"Our mandate was to deal with the issue of inside the four walls of the classroom, the school property, for safety, because you can have your background checks, you can have all kinds of side issues or gun control," Hutchinson said. But those "will not make a difference for the safety in the classroom because you've always got vulnerability there."