Mr Murdoch’s son and fellow co-chairman, Lachlan, has insisted his father did not make those remarks, but Mr Turnbull dismissed that defence.
“He would say that, wouldn’t he?” Mr Turnbull said in an interview ahead of the publication of his book on Monday.
“I only know what Kerry Stokes told me and I took a note of the discussion with Kerry at the time. And as I say at the book, I have a contemporaneous note of the discussion with Stokes.”
A spokesman for the Stokes family and Seven, Tim Allerton, said on Wednesday: “The claim made that Rupert and Kerry spoke about the leadership of Malcolm Turnbull is wrong.” In previous statements, Mr Stokes has said the “characterisation” of the conversation was wrong.
The contested exchange is central to Mr Turnbull’s account of News Corp’s attempts to destabilise his government, something he raised directly with Mr Murdoch two days before he lost the leadership.
Mr Turnbull says he told Mr Murdoch the campaign against the government was being “fired up” by News Corp newspapers and the Sky News channel at night with the aim of restoring Mr Abbott as leader.
In his book, Mr Turnbull says Mr Murdoch replied: “I think Boris [Paul Whittaker, editor of The Australian] is the only one who wants to do that.”
Mr Turnbull argues the News Corp campaign against his government was driven in part by the Murdochs’ concern that he could not be bought.
“Media barons, and many other billionaires, like politicians who are dependent on them,” he writes.
“So, while it’s easy to say that the Murdochs thought I was too liberal, at the heart of it was the fact that they knew I was my own man, and had seen that up close many times over 40 years.
“With Abbott they had a deferential prime minister they thought they controlled.”
In Mr Turnbull’s account, this deference was displayed in the way Mr Abbott and his chief of staff, Peta Credlin, favoured News Corp editors and reporters with stories, including stories about cabinet decisions that were yet to be taken by ministers.
“I wasn’t going to run my government in partnership with Rupert or Lachlan Murdoch or their editors, and I knew they’d resent that,” he says.
“A similar assessment can be made of Alan Jones, Ray Hadley and their colleagues at 2GB – in their vanity and megalomania, Jones and Hadley berate and bully politicians who don’t kowtow to them.”
In his book, Mr Turnbull warns of a weakness in the media because of the power of proprietors and the erosion in advertising revenue.
“Our media culture today is more debased than ever,” he writes.
“Traditional curated media has seen its business model smashed by the internet; Google and Facebook in particular.
“And the crazy, fact-free rage of social media has now infested what’s left of the traditional, but still very influential, media.”
Mr Turnbull said he had held “many” conversations with the Murdochs and their chief executive, Robert Thomson, to ask why they were campaigning against him.
“My point simply was: why is there this campaign against me and against my government?” Mr Turnbull said in the interview.
“It never seemed to make any sense to me, and they would always try to trivialise it.
“Rupert would often say ‘nobody watches Sky News’ or ‘The Australian doesn’t have a big readership’.
“They were prepared to pay the price of three years of Labor to get rid of Turnbull – this terribly left-wing prime minister. A lot of this defies rational analysis. And that, I think, is because a lot of the actors weren’t entirely rational.”
David Crowe is chief political correspondent for The Sydney Morning Herald and The Age.