Thursday's meeting of the BSkyB board was its first since the crisis forced News Corp (NWSA.O) to close the News of the World newspaper, drop a $12 billion bid for BSkyB and offer up James and his father Rupert to answer questions in the UK parliament.
Several shareholders had demanded he step down to avoid conflicts of interest, fearing contamination from the scandal that acquired new dimensions earlier this month when it was revealed that a murdered schoolgirl had had her phone hacked.
One of the sources said: "The role of chairman was discussed at some length," and added that the board would be monitoring external developments that might prove relevant to Murdoch's continuing chairmanship.
Murdoch did not oversee operations at the News of the World at the time the hacking occurred but did approve large payoffs to victims after he took charge of News Corp's British newspaper operations.
He was chief executive of BSkyB before moving to News Corp and is widely considered to have done a good job.
Although there had been suggestions that he could be asked to stand down temporarily while the criminal and civil investigations continued, the board is understood to have decided that his previous record not only as chairman but as chief executive deserved their continued support.
There is unlikely to be any significant change until the annual general meeting in August, however, following the company's results, which will be unveiled on Friday morning.
The board is also expected to announce details of a cash return to investors following the collapse of the bid by Rupert Murdoch's News Corp to acquire the 61% of Sky it did not already own.
This is likely to come as a special dividend or a share buyback that prevents the global media group from further increasing its stake above 39%. Any cash return to investors that did so – an ordinary share buyback for example – is likely to prove divisive given the controversy not only raised by the phone-hacking scandal but the bid itself.
News Corp was forced to withdraw its bid at the 11th hour earlier this month as the hacking scandal mounted, after unprecedented criticism by all three main political parties and the suggestion by the prime minister that the company had grown too powerful.