CONCORD, N.H. — New Hampshire’s strong Second Amendment support helped push a “Stand Your Ground” law over a governor’s veto, but a pair of bills limiting access to guns when mental health problems arise could test the state’s entrenched belief in unfettered gun rights.

The bills will be considered next year in the aftermath of deadly shootings at Sandy Hook Elementary School in Newtown, Conn., and the Washington Navy Yard. They are among perhaps a dozen or more gun control-related bills ranging from banning assault weapons and large-capacity ammunition magazines to forbidding such a ban if passed by Congress.

Gun control advocates have failed over the years to restrict gun owners’ rights to keep and bear arms, but the two bills approaching the issue from a mental health viewpoint are getting early bipartisan support.

One proposed by state Sen. David Watters, D-Dover, would add a stringent definition of people judged by a court to be mentally ill to the list of people denied the right to buy guns during background checks. New Hampshire is one of 11 states that don’t report those names to the FBI to include in the National Instant Criminal Background Check System, Watters said.

Federal law bars people who have been involuntarily committed to a mental health facility or declared mentally ill by a judge from legally buying guns from licensed dealers. States aren’t required to make the information available to the federal government or to state agencies that perform background checks.

Watters has reached out to pro-gun groups and has won backing from the National Shooting Sports Foundation, the trade association for the firearms industry based in Newtown. Jake McGuigan, the foundation’s director of state affairs, said the group is working to get states to report the names of the mentally ill to the federal government. Rather than expand the federal background check law, the existing system should be fixed, he said.

“People can continue to talk about gun control, but in the grand scheme, if the wrong people are getting firearms, gun control will not solve the problem,” he said.

State Rep. Wendy Piper, D-Enfield, filed the second bill at the request of Enfield Police Chief Richard Crate, who says state law doesn’t let police temporarily take a gun away from someone threatening suicide or harm to others.

Piper’s bill would let police ask a judge for permission to seize firearms when people pose a risk to themselves or others. A hearing would be held within 14 days to let people explain to a judge why the guns should be returned. The removal order would be good for one year.

“We can pass all kinds of gun legislation, but we really need some tools to keep guns out of their hands while they’re dealing with this short term,” Crate said.

Kenneth Norton, executive director of the New Hampshire chapter of the National Alliance on Mental Illness, also wants assurances a judge will follow clear criteria to return firearms. “Many of these may be one-time admissions. Things are not going well in life. They really don’t pose a danger to society, and, in the long run, they don’t pose a danger to themselves,” Norton said.

Former state Sen. Robert Clegg, president of Pro-Gun New Hampshire, said he wants assurances that the bills protect gun owners’ rights.

Mental health issues may be temporary and should not legally label someone after the problem is resolved, he said.

“That should not be reason for a police chief to deny a person their right (to a handgun permit) for the rest of their life. That attitude is why people don’t seek help,” he said in an email.