In a statement, Trump said Bannon has nothing to do with his presidency and has "lost his mind".
He also compared Mr. Trump to that of former President Theodore Roosevelt.
Bannon was sacked by Trump in August as the president tried to bring order to feuding factions in the White House, but the two men continued to communicate and Bannon had remained an ally.
Bannon and Breitbart will work together on a smooth and orderly transition, the news site added. He's also losing his daily show on the conservative talk station on satellite radio network SiriusXM.
A report on the Breitbart website quotes Bannon saying, "I'm proud of what the Breitbart team has accomplished in so short a period of time in building out a world-class news platform". "And those statements still stand", he said.
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Former White House chief strategist Stephen Bannon, in the book, described Donald Trump's son-in-law Jared Kushner's meeting with a Russian lawyer as "treasonous" and "unpatriotic".
Vice President Mike Pence is standing by President Trump after some have questioned his fitness for office in the fallout of the release of Michael Wolff's explosive book "Fire and Fury: Inside the Trump White House".
Neither Bannon nor a spokesperson for Breitbart immediately responded to a request for comment.
"We will always be grateful for his contributions, and what he has helped us to accomplish", Solov said.
The Wall Street Journal separately reported that Breitbart's board members were debating whether or not to oust Bannon, and last Thursday afternoon Rebekah Mercer, the conservative mega-donor who owns a stake in Breitbart, publicly rebuked Bannon in a rare public statement.