The Mail on Sunday claims that Rupert Murdoch, chief executive and chairman of News Corp, is keen for his son to join him and wanted him to make the move at the end of last year.
However, the report was contradicted by today's Daily Telegraph, citing "sources close to the Murdoch family" who dismiss it as false.
James Murdoch was appointed chief executive of Sky in 2003 amid concerns from investors that he got the job because of his father's influence.
He has won over many in the City, has led the company into broadband and high-definition television, and is believed to have conceived Sky's audacious swoop to block a possible NTL takeover of ITV by buying 17.9% of the terrestrial broadcaster.
The competition implications of that share purchase are now under investigation by the Office of Fair Trading and Ofcom, and a separate report in today's Financial Times claims Sky is lobbying chancellor Gordon Brown to head off action by the regulators.
This week, the trade and industry secretary Alistair Darling may also order Ofcom to review the public interest issues around Sky's ownership of part of ITV, but is yet to make a decision.
The timing of James Murdoch's eventual departure from Sky has long been the subject of media speculation.
The Telegraph points out that when James joined Sky, his father was asked if he was a potential successor to him at News Corp and he replied that the time to talk about such a move would be "after five years".
It omits to mention that the comment preceded the departure of the elder Murdoch brother and heir apparent Lachlan from News Corp in August 2005, which left James the sole family member still involved in his father's business.
Following's Lachlan surprise decision to quit as deputy chief operating officer he returned to Australia to set up a media company called Illyria.
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