Reports started emerging on Monday afternoon that Moyes was set to be axed - some 16 hours before he was informed at a meeting with executive vice-chairman Ed Woodward on Tuesday morning that his contract was being terminated.
The club have denied the LMA's accusation and insisted no decision had been taken at the time the reports emerged.
LMA chief executive Richard Bevan said: "The LMA is very disappointed with the nature of David's departure from Manchester United and to read extensive reports in the press, confirming David's sacking, before David himself had been spoken to officially by the club.
"Throughout his time at United, David, as he always does, has conducted himself with integrity and professionalism, values that he believes in and that have been strongly associated with the club and its rich tradition.
"It is therefore sad to see the end of David's tenure at United being handled in an unprofessional manner."
Manchester United insisted they had acted professionally and responsibly.
A United spokesman said: "We do not accept that it was handled in an unprofessional manner.
"The decision was not taken until late Monday night and was communicated face-to-face first thing on Tuesday morning.
"When reports started emerging on internet news sites on Monday afternoon, no decision had been taken at that time."
Meanwhile, Bevan added that Moyes has a "significant future" ahead of him.
He said: "David is one of the most talented, knowledgeable and dedicated managers in British football. He has a wealth of experience accumulated in a management career that started when he was 35 and already spans 16 years, with over 800 games in the professional game.
"David is a three-time winner of the LMA Manager of the Year Award, and without a doubt, he has a significant future in front of him in football management.
"He is passionate about the game and I know he possesses the drive and resilience to learn from this chapter of his career and move on to a new challenge in the game."