WASHINGTON—The White House signaled Friday that President Donald Trump is backing away from a broad overhaul of U.S. gun policy and focusing on more modest changes, after his Thursday night meeting with the National Rifle Association.

At a televised meeting earlier in the week with lawmakers, Mr. Trump had endorsed a Senate proposal to significantly expand background checks on gun sales and said he wanted to see sales of some firearms restricted to people who were 21 or older. He also suggested some guns be taken away from...