Comcast, the world's biggest entertainment group, said it had secured the backing of Sky's independent directors for a 14.75 pounds per share offer that came just 16 hours after Murdoch's 21st Century Fox (FOXA.O) had offered 14 pounds.
Hong Kong-based hedge fund Case Equity Partners, a Sky investor, said the fact Disney was in a slightly more favourable position for Fox's US media assets meant Comcast would fight even harder to get Sky.
Newly appointed Culture Secretary Jeremy Wright said he agreed with his predecessor's final decision to accept Fox's planned sale of Sky News to Disney should it succeed in buying Sky - a move clearing the final regulatory hurdle for Fox.
Regulators are concerned that Fox's ownership of Sky News would give Murdoch and his family too much influence over United Kingdom media. Comcast wants to disrupt those plans with its $34 billion offer.
Disney and Comcast are also locked in a separate $70 billion-plus battle to buy most of Fox's assets, which include the 39 percent stake of Sky it already owns and the rest of the company if it emerges triumphant in Britain.
Sky gives Fox and Comcast a rare opportunity to diversify out of the United States and reach more consumers directly.
20th Century Fox recently set July 27 as the new date for stockholders to vote on the sale of their many film and television assets to The Walt Disney Co., which has also recently received US anti-trust approval. Murdoch has a deal to sell Fox's entertainment assets including its Sky stake to Disney, though Comcast is weighing another bid for the Fox portfolio.
"Comcast has long admired Sky and believes it is an outstanding company and a great fit with Comcast", Comcast said in a statement.
Fox said the performance of Sky since 2016 justified its new bid.
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.
The pursuit for Sky is meanwhile further complicated by a battle between Walt Disney and Comcast for key assets of 21st Century Fox.