WASHINGTON—President Trump revealed his short list of candidates to be the next U.S. national security adviser, citing a number of current and former administration officials to succeed the recently departed John Bolton.

Speaking to reporters on Air Force One en route to Mountain View, Calif., Tuesday, Mr. Trump said he narrowed the list of contenders to five, for the person who will be his fourth national security adviser. Mr. Bolton left last week over differences with the president on a range of policy matters.

Mr....