Desmond bought up a million pounds woth of shares just before the proposed move the the English Nationwide league was announced and the financial services authority are inestigating a possible breach of the rules regarding "Insider" dealing.
A spokeswoman for the FSA confiemed the investigation and commented:
"The code states that directors and major shareholders shouldn't deal in shares during periods when they are in receipt of price-sensitive information."
Asked if Desmond's share purchase was being investigated, the spokeswoman said the FSA never revealed if such investigations were taking place but added: "We are looking into it "
Celtic spokesman Ian Jamieson said: "We do not comment on an individual's shareholding."