Former Celtic defender Mackay watched his side lose 3-0 at home to Southampton thanks to a double from Jay Rodriguez and a Rickie Lambert strike.
Cardiff's on-field performance added to the sense of off-field discontent caused by the ongoing battle between Mackay and Tan, which has put the manager's position in jeopardy.
The lingering uncertainty over Mackay's future, despite a statement from chairman Mehmet Dalman that he would be in charge for the "foreseeable future", fuelled a pre-match protest where a group of supporters voiced their disapproval of Tan's running of the club.
Chants of "Don't sack Malky Mackay" and "We'll always be blue" also rang around the stadium during the game.
It is understood the meeting could take place tomorrow, when the Bluebirds are due to face Sunderland, but when asked if a date had been set, Mackay said: "Not as far as I'm aware. I've not been told of any meeting. I don't usually talk to him [Tan] on a game day. That's not something that normally happens. I usually speak to other people."