Mr. Mueller is investigating Russia's meddling in the 2016 presidential election and allegations of collusion involving the Trump campaign.
The House intelligence committee has voted to release a Democratic rebuttal to a GOP memo on the Russian Federation investigation.
The standoff comes after the committee has twice delayed Bannon's return to the committee amid negotiations over whether he could answer questions about topics during the transition period and his time at the White House. Bannon is also scheduled to be questioned by Department of Justice Special Counsel Robert Mueller next week. Members of the committee have said privilege would not apply.
The House Intelligence Committee confirmed Tuesday it was giving Bannon another week to comply with its subpoena.
Gerhardt added that the degree to which the White House can exert control over his testimony before investigators depends on Bannon and his team. "Should Bannon maintain his refusal to return and testify fully to all questions, the Committee should begin contempt proceedings to compel his testimony", Schiff said in a statement. On Friday, Trump and members of the Republican party released a controversial memo concerning the probe. That assertion "stopped this last time", Rooney said.
Although executive privilege is not a new idea and has been asserted by past presidents, constitutional law professor at University of North Carolina School of Law Michael Gerhardt said only Trump himself can invoke the argument. Both Republicans and Democrats disputed that, and Democrats also bemoaned the release of formerly classified information and the possibility the precedent could compromise future investigations. Specifically, the memo takes aim at the FBI's use of information from former British spy Christopher Steele, who compiled a dossier containing allegations of ties between Trump, his associates and Russian Federation.
Trump has five days to decide whether to allow the Democratic memo's publication.
Tensions between Trump and the Democrats were high before the vote, as the president and Schiff traded insults on Twitter today - less than a week after Trump called for more bipartisanship in his State of the Union address.
Trump says on Twitter Monday: "Little Adam Schiff, who is desperate to run for higher office, is one of the biggest liars and leakers in Washington, right up there with Comey, Warner, Brennan and Clapper!"
The White House and the president's legal team have not responded to requests for comment.
The House of Representatives Intelligence Committee (oxymoron?) is about to release a totally discredited "secret memo" containing cherry-picked classified information and serious intentional omissions.