Mackay – who is worried about "seriously inaccurate and misleading reports" surrounding his conduct in the media – is at the centre of a scandal that also involves Iain Moody, the former head of recruitment at Cardiff.
The pair are alleged to have exchanged a series of insulting texts to each other, reportedly uncovered in an investigation by the Welsh club.
Moody – who resigned from his position as sporting director at Crystal Palace on Thursday – and Mackay were both sacked last season by Cardiff, who were relegated from the Premier League in 2013-14.
Cardiff filed a dossier to the Football Association (FA), alleging misconduct between the two, and the reports leaked as Mackay was seemingly on the verge of taking on the vacant managerial post at Palace.
The FA confirmed to Perform on Thursday that it had launched a probe into the claims of misconduct.
The League Managers' Association (LMA) then released a statement later on Thursday that said that Mackay had "it seems" sent text messages that were "very regrettable and disrespectful of other cultures".
The contact from Mackay, described as "friendly text message banter", came when he was under intense pressure, the LMA claimed, while it also said the Scotsman regretted his involvement.
"The LMA wish to clarify the position in relation to recent reports and speculation regarding matters alleged to have arisen during Malky Mackay’s time at Cardiff City FC," the statement read.
"In the course of a search by the club in early 2014 of 10,000 private text messages sent to and from another member of staff during Mr Mackay's employment at Cardiff, in relation to other matters, it emerged that Malky had, it seems, sent a couple of one line texts that were, with the benefit of hindsight, very regrettable and disrespectful of other cultures.
"These were two text messages sent in private at a time Malky felt under great pressure and when he was letting off steam to a friend during some friendly text message banter.
"That said, Malky believes he could and should have conducted himself better on these two isolated occasions. The precise details need to remain private for the time being until any FA process is complete.
"The LMA does not condone in any way any potential breach of equal opportunities laws but would also point out that out of over 10,000 text messages and 70,000 documents produced over a long period of time it may not be a complete surprise that some inappropriate comments can sometimes be made by employees, like Malky, working under great pressure in highly charged situations.
"If Malky has caused any offence by these two isolated matters he would, however, wish to sincerely apologise."
The statement also said Mackay found the timing of the revelations "strange" and also aimed fire at media reports.
"Malky is also very concerned about seriously inaccurate and misleading reports of his alleged involvement in these matters in the media," it continued.
"It has never been alleged that he wrote any homophobic or sexist messages and he has confirmed that he did not do so.
"Further, there are incorrect and damaging suggestions that he sent a whole host of offensive and unpleasant messages that are simply not true and which give a grossly distorted and unfair view of Malky's involvement in this matter.
"Malky looks forward to matters being put straight in due course, following any investigation of this matter."
The LMA added that Mackay could not comment on the conduct or communications made by others and that he would be "fully co-operating" with any FA investigation.
