Bannon was a close adviser during Trump’s campaign and in his first months in office, but he was fired from his White House job in August as the president sought to bring more order to his staff operations.

Mueller is investigating allegations that Russia interfered in the 2016 campaign to try to tip the vote in Trump’s favour, as well as any potential collusion by Trump’s campaign with Moscow.

Russia denies any attempt to interfere and Trump has denied any collusion.

According to a person familiar with the arrangement, the lawyer who represented Bannon for Tuesday’s House appearance, William Burck, is not representing him in connection with Mueller’s investigation. The person did not believe Bannon had hired a counsel for that yet.

According to a person familiar with the matter, Bannon doesn’t plan to assert executive privilege in his meeting with Mueller.

On Tuesday it was reported that Bannon had refused to testify to the House Intelligence Committee on Tuesday regarding conversations with key Trump administration officials.

Bannon reportedly cited executive privilege in the hearing, which is part of a second, separate investigation into Russian involvement in the election, at the behest of White House lawyer Uttam Dhillon.

Dhillon, the lawyer responsible for responding to congressional investigators in the Russia probe, made the executive privilege request to Bannon’s lawyer, William Burck, before Bannon’s testimony Tuesday said the person, who discussed the situation on condition of anonymity.

The request, which Bannon chose to honour, sparked a battle over the reach and limits of executive privilege on Tuesday as Bannon met behind closed doors with the Intelligence panel’s Russia probe.

The committee quickly issued its own subpoena for his testimony and Bannon has been invited to return to the committee as early as Thursday to answer questions around events that took place during the transition, his time at the White House, and more recently.

Press Secretary Sarah Huckabee Sanders, asked about the White House’s involvement in Bannon’s testimony Tuesday, said the administration is cooperating with the investigation.

“As with all congressional inquiries touching upon the White House, Congress must consult with the White House before obtaining confidential material,” Sanders said.

“This is part of a judicially recognised process that goes back decades. We have been fully cooperative 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.”

Representative Adam Schiff of California, the top Democrat on the Intelligence Committee, said the White House had instructed Bannon to not answer many questions on the grounds that it wanted him to preserve the president’s option to assert executive privilege later on.

“The scope of this assertion of privilege – if that’s what it is – is breathtaking,” Schiff said. “It goes well beyond anything we have seen in this investigation.”

“This was effectively a gag order by the White House preventing this witness from answering almost any question concerning his time in transition, in the administration, and many questions even after he left the administration,” he continued.

“This obviously can’t stand. We expect to have Mr. Bannon back in, we hope very soon, with a different position by the White House.”

Objections to the sweeping pre-emptive claim of potential executive privilege were echoed by Representative Trey Gowdy of South Carolina, a Republican who has generally supported the president.

The subpoena was issued “because it is the most tortured analysis of executive privilege I have ever heard of,” Gowdy, a former federal prosecutor, said on Fox News.

“Executive privilege now covers things before you become the chief executive – which is just mind-numbing and there is no legal support for it.”

House Intelligence Chairman Devin Nunes confirmed to reporters Tuesday that he authorised the congressional subpoena, and other lawmakers said he did so as Bannon declined to answer questions in nine hours of closed-door interviews on Tuesday.

Bannon had agreed to appear voluntarily before the Intelligence panel for questioning Tuesday. He was invited to return on Thursday by the committee, which has also wants unspecified documents from the ex-aide, but he has yet to confirm his attendance.

The insider said the White House’s position is unlikely to change.

Bannon’s questioning comes at the same time as his once-strong relationship with Trump appears to be frayed.

Earlier this month, Trump attacked Bannon for comments he made to Michael Wolff the author of Fire and Fury, a book highly critical of the president and his family.

They included scathing remarks about Donald Trump Jnr, the president’s eldest son, for meeting during the campaign with a Russian lawyer who was said to have damaging information on Democratic candidate Hillary Clinton.

Bannon apologised for the remarks, saying that he had actually been talking about Paul Manafort, Trump’s campaign chair, who was also at that meeting.

But Wolff said that was a lie – and whatever the truth, the claims about Bannon appeared to have burned his bridges with Trump and his supporters.

Bannon has since stepped down from Breitbart News, the “alt-right” website he used to promote Trump and his agenda, and has lost the favour of the billionaire backers who originally promoted him to Trump’s team.