On Friday, Cellino told Leeds manager Brian McDermott he was being sacked, with ex-Middlesbrough defender Gianluca Festa linked with replacing him.
GFH Capital later released a statement confirming the sale had been agreed. However, it did not go into detail about the management of the club.
The statement read: "Following recent media reports and speculation, GFH Capital would like to confirm that it has agreed to sell a 75% stake in the club to Eleonora Sport Ltd, a company owned by the Cellino family.
"They have many years experience in football and plan to invest substantially in the club including the re-acquisition of Elland Road.
"The Cellino family come to English football with an ambition to support Leeds United financially to take it to the Premier League and a belief that the club can sustain top-flight status."
Dubai-based private equity group GFH Capital purchased the club for a reported £52m from Ken Bates in December 2012.
In February last year, the group said it was looking to bring new investment into the club, but rejected a takeover bid from a consortium involving former defender Lucas Radebe last November.
Leeds are 12th in the Championship and have not won in any competition since 14 December.
Cellino, the 57-year-old owner of Italian Serie A side Cagliari, said he had "no choice" but to sack McDermott. "I want a coach for the club, not a manager," he said.
"Gianluca [Festa] is a nice guy but he wasn't in my plans before all of this. Leeds are a special club who have had difficult times. But what is beyond question is that the club's fans are very, very special."
Cellino, a 57-year-old agricultural entrepreneur, has a chequered background, having previously twice been convicted of fraud. He is known in Cagliari as ‘The King of Corn’ and in 2010 was linked with a takeover of West Ham.
Leeds might announce he has bought the club but any takeover would require him to pass the Football League’s Fit and Proper Person test which is mandatory for anyone purchasing more than 10 per cent in a club.
There was no comment from the Football League on Friday afternoon.
Meanwhile, it is also undertsood that the club's acting chief executive, Paul Hunt, has also been fired while questions remain over the role of the managing director, David Haigh, who had previously been fronting a takeover by the consortium Sport Capital, with reports claiming he had quit his position.
Haigh's consortium failed to secure a deal to purchase a 75% stake of Leeds from GFH Capital this week with Cellino coming to the fore.