The news of Mueller's action, which was first reported by the New York Times, came on the same day that the House Intelligence Committee issued a subpoena to Bannon for refusing to answer a range of questions from investigators during a combative closed-door interview, frustrating members of both parties who are probing the Trump campaign's alleged Russia ties.
"Only Steve Bannon could unite this committee," said Rep. Mike Quigley (D-Ill.).
The twin developments intensify the pressure on the former Breitbart News executive after his public falling-out with President Trump, who reacted angrily to comments Bannon made in the new book "Fire and Fury: Inside the Trump White House."
In the book by journalist Michael Wolff, Bannon criticized Trump for his firing of FBI Director James B. Comey and said it was "treasonous" of Trump's son Donald Trump Jr. to accept a June 2016 meeting with a Russian lawyer who he thought would provide damaging information about Democratic candidate Hillary Clinton.
After the book's publication, Bannon expressed regret about his statements and called Trump Jr. "a patriot."
Bannon — the first top Trump adviser known to be summoned by special counsel investigators for a grand jury interview — could provide valuable testimony about what he witnessed in the final months of Trump's presidential campaign and inside the White House, where he served as chief strategist until August.
"They are going to try to get an understanding of the relationships and how things worked and explain why he came to that opinion," said Peter Zeidenberg, a former deputy special counsel.
Bannon has told associates that he did not see any evidence of collusion between the Trump campaign and the Russian government but thinks Mueller's case could focus on financial improprieties, according to two people familiar with his views.
Bannon did not respond to requests for comment. White House lawyer Ty Cobb, who has said he expects Mueller to wrap up his investigation related to the president shortly, declined to comment.
Mueller's office has typically sought voluntary interviews from current and former Trump aides but made no such request of the White House to interview Bannon, according to a person familiar with the special counsel's communications.
By sending a subpoena to Bannon before he spoke to Congress, some legal experts said they think Mueller was seeking a way to secure Bannon's full cooperation.
"This is really Mueller's way of cementing a day, time and place for that conversation — rather than negotiating an interview," said Barbara Van Gelder, a veteran white-collar defense lawyer. "If he's going to go [to a congressional committee] voluntarily, he can testify before the grand jury."
Bannon's appearance Tuesday before the House panel perplexed and frustrated investigators, according to people familiar with the session. He arrived with no documents, saying that if lawmakers wanted emails and other communications, they were likely to be available from other witnesses.
And he refused to answer questions about conversations he had with Trump after leaving the administration or communications he had with officials outside government about people serving in the White House.
Rep. Adam B. Schiff (D-Calif.) said that Bannon's attorney, William Burck, told the committee that while Bannon himself was willing to speak with lawmakers and investigators, he could not answer questions because the White House told him not to respond.
"This was effectively a gag order by the White House preventing this witness from answering almost any question concerning his time in the transition or the administration," Schiff said.
"The scope of this assertion of privilege, if that's what it is, is breathtaking. It goes well beyond anything we have seen in this investigation," he added. "If the White House is permitted to maintain that kind of gag rule on a witness, no congressional investigation could ever be effective. So this obviously can't stand."
Burck said in a statement that "executive privilege belongs to the President of the United States. It's not Mr. Bannon's right to waive it."
When asked earlier Tuesday if the White House instructed Bannon not to answer certain questions, press secretary Sarah Huckabee Sanders responded, "As with all congressional inquiries touching upon the White House, Congress must consult with the White House prior to obtaining confidential material."
"We've been cooperating fully with these ongoing investigations and encourage the committees to work with us to find an appropriate accommodation in order to ensure Congress obtains information necessary to its legitimate interests," she added.
Faced with Bannon's resistance, Rep. K. Michael Conaway (R-Tex.), who is running the panel's probe, said that the committee elected to serve him with a subpoena during the interview.
Bannon's attorney consulted with the White House after the panel issued the subpoena, but "the White House effectively doubled down on the gag rule," Schiff said.
Conaway and Schiff said the White House never formally invoked executive privilege for Bannon, and Conaway added that he was not aware of the White House invoking privilege with any witness who has appeared before the committee in its Russia probe.
Conaway also stressed that Bannon is not in a position to invoke executive privilege himself.
A person familiar with the situation said that any executive privilege asserted by the White House with regard to Bannon would not extend to an interview by Mueller, who is also an executive branch official, and that Bannon intends to cooperate fully with the special counsel's investigation.
It is not clear when Bannon might return to the House committee under subpoena, which Conaway stressed "stays in effect, and we're going to get answers."
Josh Dawsey contributed to this report.