Updated at 11:56 p.m. on Friday, March 2, 2018.

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FILE - In this Jan. 29, 2013, file photo, former Democratic Gov. Phil Bredesen speaks on a panel about reducing the federal debt at Lipscomb University in Nashville, Tenn. Democrat James Mackler says he's dropping out of the U.S. Senate race in Tennessee to replace Sen. Bob Corker, a move that appears to clear the path to the Democratic nomination for Bredesen. (AP Photo/Erik Schelzig, File)

NASHVILLE — U.S. Senate Democratic candidate Phil Bredesen said Friday he backs universal background checks to help ensure firearms don't end up in the wrong hands.

Speaking with reporters after picking up his filing petition to run for Senate, the former Tennessee governor emphasized his support of Second Amendment gun rights.

But Bredesen said that in the wake of last month's deadly mass shooting at a Parkland, Fla., high school, "I really think there are practical things that can be done to tighten up this [National Instant Criminal Background Check System] process.

"First of all, just making it universal, [and] properly funding it. So, both the response of NICS and the work that it takes to get the information into that database," Bredesen added.

The NICS is used to check to seek if persons seeking to buy firearms aren't eligible based on their criminal background or disqualifying mental health determinations.

"I really think it's important to find a way to get mental health incorporated into this " Bredesen said. "I'd like to see a part of it be a judicial process or some process where people's rights are respected."

But he noted that "there were so many people out there who thought [the Lakeland shooter] was problematical, and I think a much more robust approval system might well have kept those guns out of his hands."

Bredesen also said he's not opposed to banning the sale of bump stocks, the after-market devices that make semi-automatic weapons act fully automatic.

"That's OK, it's a little minor piece," Bredesen said. "We've always had a prohibition against fully automatic weapons, and that's just kind of a sneaky way to get something approaching a fully automatic weapon. So I would not have a problem with that."

A campaign spokeswoman for Republican U.S. Rep. Marsha Blackburn had no immediate comment on what the Brentwood Senate hopeful thinks should be done to promote safety.

Nor did the campaign offer the congresswoman's take on remarks made by Trump this week in a White House meeting she attended with a bipartisan group of lawmakers.

The meeting on gun violence and safety issues was spurred by the killing of 17 students and faculty at Parkland's Marjory Stoneman Douglas High School by a former student on Feb. 14.

During the meeting, Blackburn responded to the mental health question, saying that "we need to be looking at the tools that [officials] have" and looking at young adults and those who control their mental health records "and how law enforcement has the ability to get that information from children's services," according to a meeting transcript posted by Vox on its website.

Blackburn said in speaking with young mothers, "they have said one of the things we need to do as we review these issues is look at entertainment. And the video games, the rating system, the movies. How things are approved and what children are being exposed. And especially children that have some of these mental health issues. They feel that has a role to play."

Asked about Trump's remarks, Bredesen said, "Well, I've kind of learned with the president there's usually some good ideas and bad ideas lumped into these kinds of things. I don't know where he is today as opposed to yesterday on this."

Bredesen emphasized that "I want to do stuff to prevent this kind of thing from happening, but I really am a Second Amendment supporter. I do believe in the right for people to own guns if they want to, if they feel they need them for protection or want to use them for sporting purposes."

Moreover, Bredesen said, "I think we can find a mix in this country of protections to keep them out of the hands of the wrong people without really denying this right to the vast, vast, vast, vast majority of people who handle guns and treat them responsibly."

Bredesen and Blackburn are running for the seat being vacated by Republican Sen. Bob Corker of Chattanooga, who announced in September he would not seek re-election.

Corker had reconsidered that action in recent weeks, but earlier this week he definitively ruled out getting back into the contest.

Bredesen, a friend of Corker, said he would not have entered the race initially if Corker had decided last year to seek re-election. But he said he was prepared to run against him if Corker had gotten back in.