Mackay, 41, has previously insisted he will not resign, but Tan reportedly sent the former Watford boss a letter by email on Monday, outlining his grievances at length.
Tan is said to have criticised Mackay's transfer signings, style of play, results and performance as a manager.
A Cardiff spokesman said: "The club is not party to any letter sent from one person to another."
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 is reported to have been 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.
Tensions have been running high between the pair after Tan publicly criticised Mackay for saying he was hoping to sign three new players in the January transfer window.
Tan accused Mackay of unfairly raising the expectations of the club's supporters.
Liverpool boss Brendan Rodgers, whose team play Cardiff on Saturday, defended Mackay on Thursday morning and criticised Tan for his attitude towards the coach.
Mackay took Cardiff into the top flight at the end of last season for the first time since 1962.
So if Mackay goes, will the 23 year old work experience guy get another promotion?- dean0