President-elect Donald Trump is widening the circle of candidates for secretary of state and will interview more prospects this week, transition officials said, a sign that after multiple meetings with high-profile hopefuls he still isn’t sold on whom he wants as the nation’s top diplomat.

Though Mr. Trump’s transition team said last week that the search had narrowed to four finalists, new candidates have emerged, including Rex Tillerson, chairman and chief executive of Exxon Mobil Corp., one transition adviser said.

Mr....