The source familiar with the matter said the White House had instructed Bannon not to answer questions about his time during the transition and the White House unless and until the committee and the White House can reach agreement on the proper scope of questioning in light of executive privilege concerns.
“The scope of this assertion of privilege, if that's what it is, is breathtaking. It goes well beyond anything that we've seen in this investigation,” Schiff said. “This was effectively a gag order by the White House preventing this witness from answering almost any question."
In a statement to ABC News, Bannon attorney William Burck said, "Executive privilege belongs to the President of the United States. It’s not Mr. Bannon’s right to waive it.”
Earlier Tuesday, when asked whether the White House had told Bannon not to answer certain questions, press secretary Sarah Sanders responded, "No one has encouraged him to be anything but transparent. But there is a process of what that looks like and what that process should go through."
"Look, we've been completely cooperative throughout this entire process. We're going to continue to be cooperative. But we're also going to maintain some of the executive privileges here at the White House that have been practiced for decades and that need to be maintained," Sanders added.
The same source familiar with the matter, while confirming Mueller's subpoena of Bannon, said that the White House has not placed any restrictions on Bannon’s testimony to Mueller and that he is free to cooperate fully with Mueller’s investigation and answer any questions Mueller’s team poses to him.
Meuller's office has not responded to a request for comment from ABC News.
“Whether he was ever given any instructions that he felt might hinder the Russia investigation is of interest to us, whether he was witness to any actions to obstruct the investigation is of deep interest to us. If he's precluded from answering any questions during transition or administration and many questions even after he left the White House, obviously we're not able to do our job,” Schiff said.
Bannon was “vociferously defending Trump” behind closed doors, according to a source close to Bannon. “It got heated. He was showing his loyalty to Trump.”
Bannon not only declined to answer questions about his time in the White House and transition but also about transition emails, according to committee members.
“Unfortunately, there were a lot more non-answers than answers to many of the questions,” Rep. Joaquin Castro, D-Texas, told CNN in an interview. “ I think we'll have to have Steve Bannon back."
The book includes harsh comments from Bannon on the controversial June 2016 Trump Tower meeting between campaign officials and a Russian lawyer. Bannon, according to Wolff, called the meeting "treasonous" and "unpatriotic," but has since retracted his criticism of Donald Trump Jr.'s participation.
In the book, Bannon also suggested Special Counsel Robert Mueller's investigation would focus on money laundering, something congressional investigators hoped to question him about.
<i>ABC News' Ali Dukakis contributed to this report.