The Cardiff boss has been told his job is safe "for the foreseeable future" despite an email from Tan last week telling him to resign or be sacked.
Mackay is seeking urgent talks with Tan, but the City owner twice declined offers to meet before Christmas.
The manager's focus now is on a busy programme of games beginning with the Boxing Day visit of Southampton, but in the meantime he has asked League Managers' Association chief executive Richard Bevan to intervene.
"I was disappointed the email was leaked and also disappointed in the content of it," Mackay said. "So it's something we'll be addressing.
"And I've already started addressing that. I've got Richard Bevan of the LMA, who's a fantastic man and someone who helps out in situations like this, and he's addressing that."
Asked about the delay in meeting Tan, he said: "I suppose people have got busy lives.
"This is something that's very important to me, something I would have liked immediately, because I'd like to get Cardiff City back focused on the football and out of the headlines for the wrong reasons.
"You can't have your dirty linen washed in public every week of the year without it affecting the club.
"I'm not saying it immediately affects players because it's my job to protect them, and they're a great group of people who focus entirely on their profession.
"There's a steely determination from me and my staff to make sure everything is done to protect them and give us the best chance of staying in this league."