(CNN) - Sen. Lindsey Graham does not support extending background checks to gun sales between two individuals, nor does he think such a bill would pass the Senate, but he said Sunday he will not hold the measure up with a filibuster
"The only way I would filibuster a bill is if Sen. (Harry) Reid did not allow alternative amendments," the South Carolina Republican said on CNN's "State of the Union with Candy Crowley."
That means he won't be joining Sen. Rand Paul, R-Kentucky, on the Senate floor if the junior senator decides to protest the gun violence legislation Reid introduced this month. Reid, D-Nevada, is the Senate majority leader and proposed a bill which would expand the scope of background checks on gun purchases, add new school safety measures, and crack down on gun trafficking.
It does not include a ban on assault weapons or magazine capacity restrictions that some gun control advocates have argued would help reduce gun violence, but Reid said he would accept amendments to the bill on the Senate floor in April.
Sen. Richard Blumenthal, D-Connecticut, said on CNN he would introduce an amendment related to gun magazine sizes.
"The majority leader has assured me and other proponents of these measures that we can offer amendments on both the assault weapons ban and the prohibition on high-capacity magazines," he said. "So there will be votes and I intend to spearhead that amendment on the high-capacity magazines."
Graham explained he is opposed to the background check expansion because the federal database and system is "clearly broken" and in need of improvement.
"This idea of private individuals transferring their weapons and having to go through a background check makes no sense," he said. "I think that legislation is going nowhere. But I would like to have a robust debate about improving the system to make sure that people who are mentally ill do not get a gun, to begin with. And there's a lot we can do on a bipartisan basis."
He and three other senators - Republican Jeff Flake of Arizona, plus Democrats Mark Pryor of Arkansas and Mark Begich of Alaska - have advanced "a bill to redefine mental health adjudications" which would ensure that not only convictions for crimes, but pleas of insanity and court orders to undergo mental health treatment are included in the database. The bill has the support of the National Rifle Association, unlike many other legislative proposals.
Sen. Chuck Schumer, D-New York, predicted Sunday on NBC's "Meet the Press" that a measure expanding background checks would fare well in the Senate.
"I called it the 'sweet spot' because it would do a whole lot of good and had a good chance of passing," he said of the background check issue. "I'm working very hard with both Democrats and Republicans, pro-NRA and anti-NRA people to come up with a background check bill that will be acceptable to 60 senators and be very strong and get the job done."
"It's very hard," he allowed. "We're working hard and I'm very hopeful that we can get this passed."
But Mark Kelly, the gun control advocate who is married to former congresswoman and gunshot victim Gabrielle Giffords, does not believe an expansion of background checks should stop with the legislation Graham proposed. He argued exempting private sales from the background check requirement leaves open a significant loophole in the firearms marketplace.
"It's crazy that we have a system … that responsible gun owners get a background check, and the option to avoid one is available for anybody who doesn't want to do it," he said on "Fox News Sunday." "Any bill that does not include a universal background check is a mistake. It's the most commonsense thing we can do to prevent criminals and the mentally ill from having access to weapons."
He recently began a purchase of an AR-15 semiautomatic rifle from an Arizona gun retailer and said the background check process was simple and characterized incorrectly by opponents. The seller, however, canceled the sale after Kelly spoke out about the ease of the transaction.
"In five minutes and 36 seconds is the time it took to fill out one piece of paper," he said. "You only have to fill out one side for it to be submitted into the National Instant Criminal background system and get an answer. Five minutes and 36 seconds. So what it shows you is it is not the burden that the NRA leadership says, what a background check is."