Murdoch made his plea during a White House dinner before Trump left for vacation in New Jersey on August 4, the Times said.
The timing means it preceded the violence that broke out over the weekend at a white supremacist rally in Charlottesville, Virginia, leaving one dead and 19 wounded.
Bannon's critics label him a white supremacist.
The dinner was also attended by Jared Kushner, one of Trump's closest advisers and the husband of his daughter Ivanka, and Trump's new chief of staff John Kelly, the paper said, quoting a person familiar with the conversation.
Trump did not strongly counter Murdoch's advice and expressed frustration with Bannon, the paper said.
CBS News also said Bannon's job is in jeopardy after Trump failed to speak out immediately against white supremacists after the violence in Virginia. Bannon could be gone by the end of the week, CBS said, quoting a source it did not identify.
Bannon, the former head of ultra conservative outlet Breitbart News, is described as being the nucleus of one of several competing power centers in what has been a chaotic White House.
The Times said Kelly has warned he will not tolerate what he sees as Bannon's behind the scenes maneuvering in the West Wing.
The paper quoted a dozen current and former Trump aides and associates who have knowledge of the situation.
Murdoch, a founder of Fox News, which Trump is said to watch assiduously and often praises as he criticizes other mainstream media as sources of fake news, is close to Kushner, the Times said.
And Kushner does not get along well with Bannon, it added.
The Times said Bannon is in disfavor for allegedly leaking stories about White House colleagues who he feels do not sufficiently adhere to Bannon's populist agenda, fighting nastily with Kushner and creating a unit that operates outside the chain of command in the White House.
Eyewitness? Submit your stories now via social or:
Email: news@pulselive.co.ke
