Comcast and Walt Disney (DIS.N) are locked in a separate $70 billion-plus battle to buy most of Fox's assets, which would include Sky, and Disney is backing Murdoch in his pursuit of the British company.
Disney recently increased their Fox Studios offer from $52.4 billion to $71.3 billion following Comcast offering a $65 billion all-cash deal for Fox.
The fight for ownership of Sky is playing out against the backdrop of Disney and Comcast's battle for a raft of Fox assets, including Fox's current 39% stake in Sky.
Last month, the Justice Department gave Disney the green light to purchase Fox provided it sold off the regional networks.
New York-listed 21CF raised its offer to 14 for each outstanding share, up substantially on a previous bid of 10.75.
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Present in 23 million homes across Europe, Sky is a prized asset, with a slate of top sport and original drama content.
Fox said it had secured the agreement of the independent committee of Sky for the deal.
Fox's long-running pursuit for all of Sky has been plagued by United Kingdom government fears over media plurality and broadcasting standards and the influence of Australian-born USA citizen Murdoch.
Unfortunately, Fox's troubles are not over yet as a shareholder Robert Weiss has initiated a lawsuit in order to prevent the deal from going ahead, according to the Hollywood Reporter.
"Comcast has long admired Sky and believes it is an outstanding company and a great fit with Comcast". It's also thought that, despite welcoming a new counteroffer from Comcast, Rupert Murdoch is in favor of Disney over Comcast due to antitrust concerns, even after the successful AT&T and Time Warner merger.
The improved bid means Sky investors now have 60 days to weigh up the offers. News Corp., which is controlled by the Murdochs, withdrew its bid for Sky soon after.