The League announced on Monday that Cellino's conviction last week for a tax offence in Italy constituted "a disqualifying condition under its owners' and directors' test".
The Italian has 14 days to lodge an appeal and his lawyers have said he will do so.
GFH Capital has confirmed it remains in talks with the League and Cellino's company, Eleonora Sport, in a bid "to find a solution that is suitable to all parties".
Cellino has been bankrolling the club for the last two months and Leeds fans now fear the club may slide towards administration.
Leeds chief executive David Haigh has categorically denied that will happen, but fans' group LUST wants urgent assurances.
"Our members, and the wider fan base, are understandably concerned about the course the club will now take," LUST said in a statement.
"When GFH bought the club the fans were promised long-term stability, but little more than a year later Leeds United is again at the centre of a long-running takeover process that is overshadowing every part of the club.
"Since the end of November the news has been dominated by GFH's attempts to sell Leeds United, with the result that we find ourselves in March with no sale, and no evidence of a plan B.
"It is time for the people who own and run the club to make substantial and concrete statements about what they are going to do now.
"While David Haigh has promised in the press that the club will not go into administration, the letter that Gibson Dunn solicitors sent to the Football League last week warned that delays to Cellino's takeover could cause "potentially irreparable damage to the Club," and the fans need to be told as a matter of urgency what this means.
"Leeds United Supporters' Trust also request that GFH urgently explain their stance towards other potential buyers following the rejection of Mr Cellino's bid."
Together Leeds, a rival consortium to Cellino's headed by former Manchester United International director Mike Farnan, have been waiting to enter into talks and it is understood there are other unnamed parties ready to step up their bids.
LUST also called on the Football League "to say what they are going to do to safeguard the future of one of their members."