The lifelong Oldham fan conveyed his decision to the club's owner via a text message on Wednesday, Lemsagam said.
"I'm disappointed by Paul's decision to resign and surprised by the reasons he has given in his statement," Moroccan Lemsagam said in a statement on the club's website.
"Both myself and the staff at the club gave Paul all the support and autonomy that he asked for. I personally was always supportive of any decisions Paul made or consulted me on reassuring him that he was the man in charge.
"He did not raise any concerns at all about the team or any other matters informally or formally, prior to his resignation. He gave no opportunity or indications that he needed to address any issues at all."
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Scholes, who has been a television pundit in recent years, was the latest from England's so-called "Golden Generation" of players to move into management, following Frank Lampard at Derby County and Steven Gerrard at Scottish side Rangers.
The Premier League winner endured a bumpy start to life in his new role but Lemsagam said he was backed to succeed despite poor early results.
"I continued to encourage him despite the results and reassured him that his job was secure. I was content to remain in League Two for this season and working with the team we have for the remainder of this season is important," Lemsagam added.
"Management is a difficult job and both myself and everyone at the club would have worked with Paul to find solutions, had we been given the chance."
Scholes' assistant Mick Priest will continue to work with the team while they seek a replacement.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
