The meeting of the BSkyB board was its first since the crisis forced News Corp to close the News of the World newspaper, drop a $12bn (£7.3bn) bid for BSkyB and summons 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.