“He will temporarily be more dangerous on the outside, but he still won’t be able to get anything done on the inside, and that’s the problem,” said Roger Stone, an informal adviser to the president for decades, who has recently become deeply critical of Mr. Bannon.

Some White House veterans said Mr. Bannon should not underestimate the value of geographic proximity.

“I don’t think Bannon can have more influence on policy from the outside, whatever he does, simply because he won’t be at the table to make the case when decisions are made,” said David Axelrod, who served as President Barack Obama’s chief strategist and has written about his frustration over his diminished access to Mr. Obama after he left the White House in 2011.