The measure is being pushed by Democrats in the wake of the Paris attacks. It's similar to a bill that George W. Bush's Justice Department supported, but that never won approval due to opposition from gun rights groups.
Now, though, the GOP front-runner said on ABC's "This Week" that he'd support it.
Host George Stephanopoulos asked: "Mr. Trump, yes or no, should someone on the terror watch list be allowed to buy a gun?"
Trump responded: "If somebody is on a watch list and an enemy of state and we know it's an enemy of state, I would keep them away, absolutely."
But Carson said later on the same show that he's concerned that "there are a lot of people on that watch list and they have no idea why they're on that list and they've been trying to get their names off of it, and no one will give them information."
"I am a big supporter of the Second Amendment, and I don't want to deprive people unnecessarily of that. There needs to be better due process," Carson said.
The legislation by Sen. Dianne Feinstein, D-California, would ban individuals on the FBI's consolidated Terrorist Watchlist from buying firearms.
Those individuals, according to a Government Accountability Office report from March, were approved 91% of the time they sought to buy guns from 2004 to 2014 -- purchasing 2,043 firearms.
New Jersey has already instituted such a ban, but the state's governor, Republican presidential hopeful Chris Christie, said on CNN's "State of the Union" Sunday that he opposes a federal version.
He told host Jake Tapper: "I think these are state-by-state determinations, Jake, and they should be made state by state."
Ohio Gov. John Kasich said on NBC's "Meet the Press" that a question from host Chuck Todd was "the first I've ever heard of it."
"If we want to examine people who are on terrorist watch lists and not let them buy a gun, I mean, it's something that ought to be considered," Kasich said.