The LMA issued a statement earlier on Friday apologising for the wording of a previous statement that was released last night which described alleged text messages between Mackay and former ally Iain Moody as "friendly text message banter", despite words being used that could be deemed as sexist, racist and homophobic.
However, Cardiff, whose owner Vincent Tan reportedly provided the Football Association with a dossier that included the text messages, have released their own statement which describes the position of Bevan as "untenable".
The statement read: "We had not intended to comment on matters which have been reported in the press over the last two days in relation to Malky Mackay and Iain Moody. However, following the statement released by the LMA last night we feel compelled to respond.
"The LMA is the representative voice of managers, and whilst we understand it seeks to act in the best interests of its members, one of its major aims is to 'encourage honourable practice, conduct and courtesy in all professional activity'.
"Regrettably, we feel that the LMA has done no such thing in its representation of Mr Mackay and Mr Moody. The LMA were aware of the allegations from the morning of the search order, and their lawyers acted for Mr Mackay and Mr Moody throughout the legal process.
"That the LMA has sought to criticise the club for the timing of the report to the FA is preposterous, because the offensive communications have been in the knowledge and possession of the LMA for many months. When the messages came to light, over three months ago, the club strongly encouraged and advised Mr Moody and Mr Mackay to deal with the issue directly with the FA.
"It was made clear to them, and their LMA appointed lawyers, that the nature of the communications meant the club was under a duty to report their findings to the FA if they did not take appropriate action themselves. Nevertheless, with the backing of the LMA, Mr Moody and Mr Mackay chose to do nothing. The LMA were therefore complicit in the attempt to conceal these messages (of which there were many more than the two isolated texts acknowledged by Mr Mackay).
"We also point out that the LMA members have a 'Code of Conduct' incorporated into their managerial contracts. One of the requirements is "A Manager shall not use racist or other discriminatory language. A manager's behaviour should demonstrate to players and other employees under his control that discrimination in any form is unacceptable."
"We therefore find it entirely reprehensible that the LMA should itself put out a statement which seeks to dismiss deeply offensive racist comments as 'friendly banter'.
"If that is the view held by the LMA, as appears from its statement, we consider that Richard Bevan's position is untenable and we call for his resignation."
Mackay had been expected to be appointed as the new manager of Crystal Palace before the text messages came to light on Wednesday evening.