Mr. Huntsman, who had at one point been considered as a candidate for secretary of state, has also been critical of Mr. Trump. Mr. Huntsman didn’t immediately respond to a request for comment.

He endorsed Mr. Trump for president in September, but in October called for Mr. Trump to drop out of the race after a recording leaked of Mr. Trump making sexually aggressive comments about women.

“In a campaign cycle that has been nothing but a race to the bottom—at such a critical moment for our nation—and with so many who have tried to be respectful of a record primary vote, the time has come for Governor Pence to lead the ticket,” Mr. Huntsman told the Salt Lake Tribune last October.