Fox raised its bid to 14 pounds ($18.58) a share as it seeks the 61 percent of Sky not already under its control.
Fox could face further competition from the United States cable giant Comcast, however, after it forced Sky's independent directors to drop their previous backing for Fox with an unexpected £12.50-a-share bid for the group in February.
But Disney and Fox are determined to keep their deal together - Murdoch has favored Disney in negotiations over the sale of 21st Century Fox assets, including Fox's stake in Sky, to Disney.
21st Century Fox has increased its bid to acquire the 61% of the hotly-contested Sky that it does not now own.
England nowhere near their potential, says manager Gareth Southgate
Southgate's side have been buoyed by more than just an anthemic pop song, though. Against Sweden (2-0 win in the quarter-finals) they were magnificent.
Disney now has a $71.3 billion deal in place with Fox to acquire the company's film production businesses, including Twentieth Century Fox, Fox Searchlight Pictures and Fox 2000 Pictures; Fox's television creative units Twentieth Century Fox Television, FX Productions and Fox21; FX Networks; National Geographic Partners; Fox Networks Group International; Star India; and Fox's interests in Hulu, Sky plc and Tata Sky.
Comcast's first £12.50 a share offer, valuing Sky at £22bn, resulted in Sky's independent committee recommending shareholders reject Murdoch's original £10.75 deal tabled in December 2016.
Disney secured conditional USA approval to buy the assets last month, giving it an edge over Comcast's bid.
"This transformative transaction will position Sky so that it can continue to compete within an environment that now includes some of the largest companies in the world", Fox said. Of the assets he's seeking to acquire from Fox, Disney chief executive officer Bob Iger called Sky "a real crown jewel". Wright's predecessor had previously indicated the deal could go through if Sky was ultimately sold to a third party, like Disney.
"American consumers have benefitted from head-to-head competition between Disney and Fox's cable sports programming that ultimately has prevented cable television subscription prices from rising even higher", said Makan Delrahim, assistant attorney general and head of the Justice Department's Antitrust Division. The opposition has not completely subsided despite the plan to spin off Sky News.
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