WASHINGTON—President Donald Trump will soon sign a revised executive order banning certain travelers from entering the U.S., but unlike the original version, it is likely to apply only to future visa applicants from targeted countries, according to people familiar with the planning.

Mr. Trump had been scheduled to sign the new order Wednesday, multiple officials said, but the event was postponed. It was unclear why the timing was changed but one person familiar with the planning said the White House was concerned about competing...