PALM BEACH, Fla.—President Donald Trump said there would be a “bright path” available to North Korea if it successfully shut down its nuclear weapons program, but he vowed he would walk out of a planned meeting with the country’s leader Kim Jong Un if he felt it wasn’t yielding results.

Mr. Trump’s remarks, made at a news conference at his Florida resort Mar-a-Lago with Japanese Prime Minister Shinzo Abe, came as U.S. officials have been negotiating the terms of a North Korea summit with Mr. Kim that the president said he...