Bahrain-based investment firm GFH Capital has written to the the governing body calling for an end to the uncertainty at Elland Road after what it described as an "unprecedented delay".
Italian businessman Cellino exchanged contracts with GFH Capital on the purchase of 75 per cent of the club's shares at the beginning of February and has been waiting for the League's approval.
But Cellino's bid to take control was dealt a huge blow on Tuesday after he was found guilty in a Sardinia court of tax evasion and it appears he will now fail to pass the League's 'owners and directors' test.
A spokesman for GFH told the Yorkshire Evening Post: "Leeds United have (on Thursday morning) sent a letter to the Football League, demanding that the Football League hands out its decision by the close of play today.
"This is an unprecedented delay which is damaging the owners' ability to move the club forward."
Leeds' managing director David Haigh added: "I've been in regular contact with the Football League throughout the whole process and I can say that both (chief executive) Shaun Harvey and other League officials have been extremely helpful every step of the way.
"Shaun has even taken calls at 3am local time!"
Cellino's lawyers confirmed after the brief court case in Sardinia that he will be appealing against the conviction in relation to the non-payment of taxes on his luxury yacht - a process that could take between nine and 12 months.
GFH Capital has been searching for major investment in Leeds since buying the club from previous owner Ken Bates in December 2012.
- Sports & Recreation
- GFH Capital
- Leeds United
- Football League