"I would certainly implement that, absolutely," Trump told NBC News at a campaign stop in Iowa on Thursday.
But on Friday, Trump asserted in a tweet that he never suggested such a database:
(Trump did not address the issue in his only public event of the day in South Carolina).
However, the Republican frontrunner told Yahoo News earlier Thursday that he would consider creating one.
"We’re going to have to -- we’re going to have to look at a lot of things very closely,” he said. "We’re going to have to look at the mosques. We’re going to have to look very, very carefully."
Trump told NBC that he wouldn't stop at just creating a database, though he didn't specify what other systems he'd like to implement.
"There should be a lot of systems, beyond databases," he added. "We should have a lot of systems."
"I find it abhorrent that Donald Trump is suggesting that we register people,” Bush said in an interview Friday on CNBC. "That haunts back to a time that no one wants to go back to."
Trump has previously said he would shut down some mosques in the United States in the wake of the terrorist attacks on Paris if deemed necessary.
"There’s absolutely no choice,” Trump told FOX News' Sean Hannity this week of the possibility of closing mosques. “Some really bad things are happening and they’re happening fast. Certainly a lot faster than our president understands because he doesn’t understand anything."