Fox has been trying to get approval from United Kingdom regulators since 2016 to buy the 61% of Sky it does not already own.
Fox made the new offer in an attempt to outbid its rival Comcast, which has offered Sky £12.5 (Rs. 1,139) per share, CNBC reported. Comcast has also offered guaranteed funding for Sky News and other commitments, including maintaining Sky's West London base.
"We could be seeing a bidding prioritization by Comcast for Sky over Fox, perhaps due to the fact that Comcast cannot afford both, and has to choose its battles".
Only this time, it's not Comcast standing in the Mouse House's way, but a shareholder named Robert Weiss, who recently filed a lawsuit to stop the historic Disney-Fox merger. The new bid values Sky at $32.5 billion.
A successful acquisition of Sky would extend Comcast's reach Europe.
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Sky's independent directors accepted the latest Fox offer.
The Government had been concerned over the impact that Fox's takeover of Sky could have on United Kingdom media, given that the Murdoch family also owns News Corp - the publisher of a raft of newspapers including The Sun and The Times.
Fox will be hoping to see off competition from Comcast after a long battle to buy the remaining stake in Sky, having faced heavy scrutiny from the Government and the competition watchdog.
The improved bid means Sky investors now have 60 days to weigh up the offers. The European pay-TV company has become a pawn in a wider contest between Comcast and Walt Disney (dis) for the bulk of Murdoch's media empire, as each seeks scale to take on streaming competitors like Netflix.
Fox shareholders are scheduled to vote on the Disney deal on July 27.
Murdoch had previously tried to buy Sky in 2011 when a phone hacking scandal at his News of the World tabloid sparked a political backlash over his role in Britain.