Two other Trump associates, White House Deputy Chief of Staff Rick Dearborn, and former campaign manager Corey Lewandowski, were due to appear before the House panel behind closed doors on Wednesday, congressional sources said.
The panel also subpoenaed Bannon for testimony as part of its Russian Federation investigation after he refused Tuesday to answer questions about his time working for Trump during the transition and in the White House.
After Mr Bannon initially refused to answer questions about his time at the White House on Tuesday, the committee's Republican chairman Devin Nunes authorised the issuing of the subpoena to press Mr Bannon to respond.
As of Wednesday morning, neither Bannon nor anyone from Mueller's office had commented on the agreement.
"This was effectively a gag order by the White House", Schiff said, adding that Bannon would likely be back in front of the committee soon, and this time without any restrictions.
"There were an very bad lot of questions we weren't able to answer based on this novel theory of executive privilege", he said.
Trump takes to Twitter to blast Democrats over DACA
Dick Durbin, a Democrat Senator, was the first to condemn Trump for his racial rant during a White House Meeting last week . The Justice Department said in a statement it "looks forward to vindicating its position in further litigation".
Bannon's statement did little to soothe tensions with the White House and the former aide was let go from his position as executive chair of the far-right site Breitbart News on January 9.
Mr Bannon apologised a few days later, but was stripped of his job leading the pro-Trump news site Breitbart News.
The special counsel investigation has so far netted indictments against four former members of Trump's inner circle during the 2016 presidential election.
Bannon also reportedly said in Wolff's book that Donald Trump Jr. personally walked Russian officials to Trump's office in NY.
"Although I may think that something might have gone on between (the Trump team) and Russian Federation, I don't think it was the deciding factor in the final vote", junior criminal justice major Mike Best said.