Bannon made the scathing comments in a book to be published next week -- the "Fire and Fury: Inside the Trump White House," by journalist Michael Wolff, the newspaper said.
"They're going to crack Don Junior like an egg on national TV," Bannon reportedly said.
Bannon, who left the White House in August, was also quoted as saying that the investigation by special counsel Robert Mueller into Russian interference in the 2016 presidential election will focus on money laundering.
Besides Donald Trump Jr, Trump's son-in-law Jared Kushner and then campaign chairman Paul Manafort attended the meeting at Trump Tower in New York.
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Donald Trump Jr took the meeting with the Russian lawyer after an intermediary promised material that would incriminate Trump's Democratic rival Hillary Clinton.
"The three senior guys in the campaign thought it was a good idea to meet with a foreign government inside Trump Tower in the conference room on the 25th floor -- with no lawyers," Bannon was quoted as saying in the book. "They didn't have any lawyers.
The investigation by Mueller, a former FBI director, is looking into whether the Trump campaign colluded with Russia to get him elected -- a charge the president has repeatedly and vehemently denied.
Trump said in an interview with The New York Times last week that he expected the Mueller probe to be "fair" and Donald Trump Jr has denied any wrongdoing.
Manafort and a business associate have been indicted on money laundering charges unrelated to the election campaign. Trump's former national security adviser Mike Flynn has pleaded guilty to lying to the FBI about his contacts with Russians.
"They're sitting on a beach trying to stop a Category Five," he said.
Bannon's relations with Kushner were reportedly strained during their tenure at the White House.
And Bannon had particularly harsh comments in the book about the wealthy New York real estate developer who is married to Trump's daughter Ivanka and serves as a senior adviser to the president.
"You realize where this is going," Bannon said of the probe by special counsel Mueller. "This is all about money laundering.
"It goes through Deutsche Bank and all the Kushner shit... The Kushner shit is greasy. They're going to go right through that."
Since leaving the White House in August, Bannon has staked out some positions at odds with the Trump administration.
In the most notable break, he defied the president and backed a rival Republican candidate in a Senate race in the southern state of Alabama. Trump eventually changed his position and also supported Roy Moore, who lost.
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