Sky Sports says the club's chairman Abdallah Lemsagam received the message out of the blue on Wednesday afternoon.
Manchester United legend Scholes has not spoken to anyone on the board of directors since sending his resignation message.
"Oldham Athletic are disappointed by Paul's decision to resign and surprised by the reasons he has given," Lemsagam told Sky Sports News.
"The club gave Paul all the support and autonomy that he asked for. He did not raise his concerns at all, informally or formally, prior to his resignation. He gave no opportunity to address any issues at all, simply resigning by text and then refusing to discuss matters.
"Management is a difficult job and we would have worked with Paul to find solutions, had we been given the chance. We wish him well and thank him for his contribution. We will start the search for a new manager immediately."
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