WASHINGTON—Former New York Mayor Rudy Giuliani is the leading candidate to be President-elect Donald Trump’s nominee for secretary of state, people familiar with the matter said, a move that would elevate a well-known national figure to become the U.S.’s chief diplomat.

Mr. Trump’s aides have also considered former U.S. Ambassador to the United Nations John Bolton as a possible candidate, but the close relationship between Messrs. Giuliani and Trump was a major consideration, the people said.

Asked...