He has criticized Kushner, Trump's son-in-law and senior adviser, for his more politically moderate approach.
Thick with tension, the conversation this week between Stephen K. Bannon, the chief White House strategist, and Jared Kushner, the president's son-in-law and senior adviser, had deteriorated to the point of breakdown.
The Wall Street Journal also reported on Friday that Trump is considering replacing Priebus and Bannon and is "trying out different names with his friends", according to someone "close to the White House".
"Reince is chief of staff", said a source.
The news comes after The New York Times reported over the weekend that Kushner had complained to President Donald Trump about the negative coverage he was receiving from the far-right website. "And he does that", asserting "He asks a broad range of people who he trusts, and Steve Bannon's one of them, about policy decisions and about the risks and opportunities involved with each of these and so none of that has changed".More news: Chelsea fans marvel in Marcos Alonso's stunning free-kick at Bournemouth
Rep. Ileana Ros-Lehtinen said in a local radio interview that it would be "welcome news for the nation" if Bannon left the Trump administration. Administration officials said this was done at the urging of national security adviser HR McMaster, with whom Mr Bannon had clashed.
He has also played a key role in Mr Trump's foreign policy, including the U.S. relationship with China.
Conservatives are an endangered species in your White House. He's channeled the populist and nationalist sentiment that propelled Trump's campaign, and his placement on the National Security Council committee was criticized by some members of Congress and Washington's foreign policy establishment, who said it risked politicizing the security advice provided to the president.
"I think we've had one of the most successful 13 weeks in the history of the presidency", he said.
Trump himself - who has wielded a management style based on staff divisions and chaos - downplayed talk of a staff shakeup on Thursday. According to sources speaking to Axios, President Trump is considering cleaning house on his staff, which entails getting rid of the controversial chief strategist and chief of staff. "Things are happening", an aide said,"but it's very unclear the president's willing to pull that trigger".