Fox's new bid values the whole of Sky at £24.5 billion, beating an offer of £22 billion from United States (US) cable giant Comcast for the satellite TV group.
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.
However, Comcast's new offer, more than 5% over Murdoch's latest deal, has won the recommendation of Sky's independent committee.
Hedge funds including Elliott have bought into Sky in recent months and other vocal shareholders such as Crispin Odey have demanded that the independent directors secure a better deal.
British pay-TV group Sky said Wednesday it accepted a sharply increased offer from 21st Century Fox to acquire the pan-European company, according to the Financial Times.
Comcast may instead try to get the Sky network, which Fox now owns 39% of the company.
This all part of the ongoing battle for Fox's entertainment assets. gets more complicated as the days go on. Sky's shares closed at 15.01 pounds on Tuesday, implying shareholders think the battle is not yet over.
Brian Roberts, chief executive of Comcast, said: "We have long admired Sky, which we believe is an outstanding company and a great fit with Comcast".
The complaint alleges that Fox's prospectus filed to the Securities and Exchange Commission after the company accepted Disney's offer was incomplete. Comcast faces a deadline of Friday to put its offer to Sky shareholders.
Fox said Disney had given its consent to the additional debt Fox would take on and said that it would reimburse Fox by around 1 billion pounds if Fox succeeds in buying Sky at that price, but the Disney-Fox transaction falls through.
The U.K. government, which has already said it's minded to approve Fox's bid for Sky after Fox satisfied concerns over media plurality, is due to deliver its final ruling on Thursday. The media mogul's previous effort to buy out the rest of Sky foundered amid Britain's 2011 phone-hacking scandal, which hardened opposition to Murdoch's influence over the United Kingdom media landscape.