Fox, which owns 39 percent of Sky and is due to gain British regulatory clearance for the deal this week, upped its offer to 14 pounds per share, from its earlier 10.75 pounds per share.
Fox could face further competition from the USA 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. Sky's shares closed at 15.01 pounds on Tuesday, meaning shareholders may also not like the price.
Fox said it had secured the agreement of the independent committee of Sky for the deal.
Present in 23 million homes across Europe, Sky is a prized asset, with a slate of top sport and original drama content.
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". According to some shareholders, that has set an implied higher floor for Sky's shares.More news: British woman dies after Novichok nerve agent poisoning
The move by the NBC owner came less than 24 hours after 21st Century Fox increased its offer to £24.5bn.
The suit aims to have the Fox-Disney merger enjoined on a preliminary basis, but if that doesn't occur, the shareholder would like to see the deal rescinded with an award of rescissory damages.
Murdoch is aiming to takeover the 61% of Sky he doesn't already own through 21st Century Fox, which owns assets including the Hollywood studio behind films including X-Men and Deadpool.
"Fox coming back in for Sky isn't a surprise in itself, but the fact the offer is slightly behind what some had anticipated brings another twist", said George Salmon, equity analyst at Hargreaves Lansdown.
The battle for Sky intensified on Wednesday night as Comcast raised its bid for Sky, valuing the pay TV giant at £26bn.
"There are enough sub-plots in the race to acquire Sky to commission a primetime drama", Salmon said.