As Senate Democrats push ahead with a proposed ban on assault weapons and other gun-control legislation, Republicans are still trying to draw attention to what they see as the bigger issue -- keeping the mentally ill from owning firearms.

A proposal on the issue was introduced this month by South Carolina Republican Sen. Lindsey Graham, a member of the Senate Judiciary Committee, who on Thursday again expressed his interest in getting the measure passed.

“I believe that the best way to interrupt the shooter is to have a mental health system that actually records and enters into the database people who should not be able to buy a gun,” Graham said.

He made his remark while voting against a bill passed by the committee to ban assault weapons and high-capacity gun magazines.

Graham’s proposal would require that people found mentally incompetent be added to the National Instant Criminal Background System – the database for all new gun sales.

Those added to the list would have to be found incompetent by a federal court or other official body. The proposal is based on a 2005 South Carolina case in which a women pleaded not guilty by reason of insanity for threatening to kill President George W. Bush. The woman, Alice Bolan, was later able to legally buy a gun and attempted to shoot somebody earlier this year.

“The … case is Exhibit A of a broken background check system,” Graham said. “As astonishing as it sounds, that actually happened.”

Graham’s proposal was introduced earlier this month and has bipartisan support.

"I’m a strong supporter of Second Amendment rights,” Arkansas Democrat Rep. Mark Pryor said in support of the bill. “That said, it’s evident that our background check system needs some improvements.”

Under the proposal, a person would be added to the database for such reasons as being an imminent danger to themselves or others, found guilty but mentally ill in a criminal case, found guilty by reason of insanity, found incompetent to stand trial, committed to a psychiatric hospital or required to have psychiatric outpatient treatment.

Meanwhile, several states are addressing the issue.

New York passed legislation that requires therapists to report to the state cases of potentially dangerous behavior. And the Maryland and Florida legislatures are considering similar legislation.

The Florida proposal attempts to close a loophole that allows the mentally ill to voluntarily commit themselves to treatment to avoid getting added to a database. The bill also would require the mandatory reporting of mental illness.

The U.S. Senate committee has now voted in favor of four gun-control measures: the assault weapons ban, more school safety aid, expanding federal background checks on potential gun buyers and helping authorities prosecute illegal gun traffickers.

However, the ban on assault weapons and high-capacity magazines faces tough odds in passing the full Senate. The measure would need 60 votes, which would require support from Republicans and some Democrats seeking re-election in pro-gun districts.