The scheduled meeting had taken on extra significance after Mackay's head of recruitment Iain Moody was removed from his post last week and replaced by 23-year-old Kazakhstani Alisher Apsalyamov.
Apsalyamov is thought to be a friend of Cardiff owner Vincent Tan's son, and has no known experience of top class football.
Chief executive Simon Lim was among those in attendance at Monday's proceedings, although Tan was not, but Press Association Sport understands talks during the meeting were largely positive and productive.
It had been clear from the outset that Mackay would not resign from his post, and the club denied he has been asked to do so.
But he was understood to be very disappointed by the decision to get rid of Moody, who he had worked with since his time at Watford, as well as being concerned at what it would mean for the future of the club.
And Mackay's agent Raymond Sparkes reiterated the Scot would not quit the Bluebirds.
When asked if Mackay would step down, he told BBC Radio Wales: "No, absolutely categorically not. That's not something Malky is contemplating.
"There is absolutely no prospect of Malky resigning from a job that he loves, a job he's ensconced in."
But Sparkes hinted Mackay does not have the full backing of everybody at the club and his future is out of his hands.
"Malky loves being at Cardiff City, has started a job there and feels that he's been given the opportunity to kind of be the architect in some respects of a new Cardiff City," he said.
"But football is a precarious business and Malky's decision to stay might not thereafter be matched by all the powers that be deciding that he should stay, so in some respects that's pretty much in the hands of other people."
On Moody's exit he added: "Both men (Mackay and Tan) have made their positions quite clear but Iain has still been removed from his position and that's not going to change.
"The real question is how do we go forward from here, how do we get things back on to an even keel and not only have a plan for taking the club forward but convince the Cardiff City fans that it's something that they can absolutely rely on and trust in going forward.
"Iain was a crucial member of Malky's team and for anyone who knows Malky or has come into contact with him, he is a tremendous character in every respect.
"No one will be surprised to hear that when a member of his family, if that's what we can call him, has been removed, then Malky on complete morality grounds is always going to jump to the defence of that individual."