A senior administration official insisted that the White House had not told Mr. Bannon to exert executive privilege.
Mr. Bannon did not address reporters, and a spokesman for Mr. Mueller did not respond to messages seeking comment.
After excerpts from the book, “Fire and Fury: Inside the Trump White House,” were published this month, Mr. Trump derided Mr. Bannon publicly and threatened to sue him for defamation.
Mr. Bannon was soon ousted as the executive chairman of the hard-right website Breitbart News.
Some legal experts said the subpoena could be a sign that the investigation was intensifying, while others said it might simply be a negotiating tactic to persuade Mr. Bannon to cooperate with the investigation.
Mr. Bannon also helped run the transition after Chris Christie, now the former governor of New Jersey, was fired as head of that team.
Mr. Bannon did not apologize, however, and though he had approved the statement, an associate sent it to reporters without his knowledge. »