Mackay, who left Cardiff under a cloud last year following a falling-out with club owner Tan, apologised on Friday after media published details of texts he sent while in charge at the Welsh club, who were relegated from the Premier League last season.
But he now faces fresh claims, printed in the Daily Mail, which represent more accusations of racism and may be seen to explain his apology, along with former head of recruitment Iain Moody back in May, for "any offence" they might have caused the Malaysian billionaire.
The Mail published a text message it claimed Mackay sent to head of recruitment Moody in June last year about a signing Tan was reluctant to sanction.
It read: "Not taking no for an answer from the chink until the 20th time or unless somebody big blows us out of the water. Can you tell him that."
In providing context, the newspaper added that Tan controlled Cardiff's finances at the time and that the club's investigators believe that the text message referred to the owner. If genuine, it was among tens of thousands of texts and emails seized in March during a dawn raid on Moody's home by investigators commissioned by Tan.
Daily Mail headline on Mackay allegations (Screenshot)
Before Saturday's report emerged, the 42-year-old Scot apologised for what he said was three "unacceptable" text messages on Sky Sports News.
"I've been in a multi-cultural football environment for 20 years. I love British football and I am no racist. I am no sexist. I am no homophobe and I'm not anti-semitic," he said in a TV interview.
"I sincerely apologise. It was something I did, and there's no excuse for that. It was in a period (when) I was under immense pressure and stress in terms of the relationships that were possibly not going too well at my football club at the time."
An apologetic Malky Mackay says his discriminatory texts were 'unacceptable and inappropriate'
The text messages under scrutiny were exchanged between Mackay and Cardiff's then head of recruitment, Moody, according to the Daily Mail which broke the story.
Mackay said he was responsible for three offensive messages.
"Obviously, it was someone else's phone that a vast array, over 10,000, of private texts were lifted from.
"I received some (of these texts) but as I say it's the three texts I'm accountable for and that I shouldn't have sent."
Former Scotland defender Mackay was sacked by Cardiff midway through last season after a public disagreement over transfer spending.
He had been expected to take charge of top-flight Crystal Palace this week but was ruled out of the running after details of the messages emerged.
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