Press Association Sport understands Tan sent Mackay a letter by email on Monday, outlining his grievances with the former Watford boss at great length.
Tan was highly critical of Mackay's signings, style of play, results and performance as a manager.
The letter also included the ultimatum that the Scot quit or be given the boot by the controversial Malaysian businessman.
A Cardiff spokesman told Press Association Sport: ''The club is not party to any letter sent from one person to another.''
Mackay, who led Cardiff to the 2012 League Cup final before securing the Championship title last term, has previously stated he would not resign from his position, and it is believed it could cost Tan somewhere in the region of £3million to sack the 41-year-old, whose contract runs until 2016.
Mackay's position as Bluebirds boss has been under scrutiny since Tan removed his head of recruitment Iain Moody in October amid suggestions of a summer transfer window overspend, something the Scot and Moody denied outright.
The problems at the club deepened on Monday, the same day as Tan's email was sent, when chief executive Simon Lim released a statement claiming Tan was ''extremely upset'' at Mackay's comments he hoped to bring in three players in January, and that there would not be ''a single penny'' available to spend next month.
Mackay is scheduled to hold his pre-match press conference on Friday morning, ahead of Saturday's Barclays Premier League trip to Liverpool, with his future set to again be the central topic.
The latest row with Tan has also seen odds on the Scot becoming the next West Brom manager slashed.