WASHINGTON — Ben Carson, the retired neurosurgeon who ran an outsider’s campaign for the Republican presidential nomination and later backed Donald J. Trump, is likely to be chosen as secretary of housing and urban development, according to a person close to the Trump transition, suggesting that the president-elect is seeking diversity in an administration that has started off predominantly white and male.

Mr. Carson would be the first African-American member of Mr. Trump’s cabinet — and a familiar face to millions of Americans after a Republican primary campaign in which he briefly soared to the top of the polls.

Mr. Trump has begun meeting with a wider array of prospects, either for advice or as potential cabinet picks. He is also considering Harold Ford Jr., an African-American former Democratic representative from Tennessee, for transportation secretary or another post, though that process has not yet become serious, a Trump official said.

Interviewed on Tuesday on the Fox News program “Your World With Neil Cavuto,” Mr. Carson said the president-elect had offered him multiple positions in his cabinet. Asked specifically about the housing job, he replied, “I would say that it was one of the offers on the table.”