Cellino's bid to buy 75% of the Yorkshire club from GFH Capital is currently being reviewed by the League to see if he passes the owners' and directors' test.
In 2001, he received a suspended sentence for false accounting, while a 1996 fine for fraud was overturned on appeal.
According to the Yorkshire Evening Post, Cellino's lawyers have argued that both convictions should be viewed as 'spent' and not affect the League's decision.
On Friday, it was reported that Cellino's company, Eleonora Sport Ltd, had agreed to buy a majority share in the football club for £25m from GFH.
With the Italian's deal yet to be sewn up, GFH are now said to be considering an offer from another consortium.