WASHINGTON—Special counsel Robert Mueller is examining what role, if any, former national security adviser Mike Flynn may have played in a private effort to obtain Hillary Clinton’s emails from Russian hackers, according to people familiar with the matter.

The effort to seek out hackers who were believed to have stolen Mrs. Clinton’s emails, first reported by The Wall Street Journal, was led by a longtime Republican activist, Peter W. Smith. In correspondence and conversations with his colleagues, Mr. Smith portrayed Mr. Flynn...