WASHINGTON—President Donald Trump is consulting with advisers about whom he should tap as his next chief of staff, with John Kelly expected to depart the administration as early as this summer, possibly soon, according to people familiar with the matter.

The two front-runners for the job, the people said, are Nick Ayers, who serves as chief of staff to Vice President Mike Pence, and Mick Mulvaney, who heads the Office of Management and Budget as well as serving as acting director of the Consumer Financial Protection Bureau.

Mr. Mulvaney dined with the president Wednesday evening, and Mr. Trump discussed the chief of staff position with him, according to a person familiar with the conversation.

Plans for Mr. Kelly’s exit aren’t final, the people said. Other aides have fallen out of favor with the president yet held on to jobs longer than expected as Mr. Trump gets more feedback on their performance.

The White House, asked about Mr. Kelly’s status, initially declined to comment. Later Thursday, after The Wall Street Journal report appeared online, White House spokeswoman Lindsay Walters said the president called the article “fake news” and that Mr. Kelly said “this was news to him.”