President-elect Donald Trump said he will nominate retired neurosurgeon Ben Carson as secretary of the U.S. Department of Housing and Urban Development, a move that would place a former political adversary with little housing-policy expertise in a key administration post.

When Mr. Carson emerged as Mr. Trump’s top rival last year during the early stages of the Republican primaries, Mr. Trump sought to discredit his personal story of emerging from poverty during his youth. After Mr. Carson faded in the polls, he moved to support...