The White House has been lobbying aggressively to maintain the authority, against a bipartisan coalition of lawmakers - conservative Republicans and liberal Democrats - who say the act compromises civil liberties.
Mr Bannon has tried to make amends.
Burck also represents White House Counsel Don McGahn and former White House chief of staff Reince Priebus in connection with the Russian Federation probe.
The committee is investigating Russian interference in the 2016 presidential election as well as allegations of collusion between members of the Trump campaign and Russian officials.
Inside the White House, Bannon was viewed as the keeper of Trump's nationalistic flame, charting the progress on the president's promises to his base on dry erase boards in his office.
After resuming his position at Breitbart, Bannon set out to challenge Republican incumbents deemed insufficiently loyal to the Trump administration.
Beyond revealing the breakup itself, Mr. Wolff's book provides a stinging insight into what some White House insiders thought of the president.
Mr Bannon could attempt to start a new venture without Mr Ailes or perhaps go back to making documentaries. One fired campaign aide, Sam Nunberg, is quoted as saying he was assigned to tutor Mr. Trump on the Constitution.
Also, a former deputy chief of staff, Katie Walsh, is quoted as saying that trying to deal with the president was "like trying to figure out what a child wants" - an observation she later denied having made.
President Trump perplexed political pundits when he boasted Thursday that he has "a very good relationship" with North Korean dictator Kim Jong Un, while in the same breath deriding his former right hand man Stephen Bannon as a traitor. The leading national reporters covering the White House, including at least one who still supports Trump, are arguing whether Trump was a moron (as stated by his secretary of state) or was "only" suffering a mental disorder that has now grown much worse. As if we didn't already know that greed trumped security.