The tempestuous relationship between the two was brought into the public eye on Monday when Tan hit out at Mackay via a statement from club chief executive Simon Lim, who said the owner was "extremely upset" that the manager had gone public with his desire to strengthen his squad in January.
The BBC now claims that on the same day that statement was released, Tan emailed a second letter to Mackay personally telling him his days at the club were numbered.
"Tan criticised 41-year-old Mackay at length in a range of areas, such as signings, transfer budgets, results on the pitch and style of play," said the BBC.
"The Malaysian businessman also questioned the former Watford manager's record as a boss."
Mackay, whose contract at the Welsh club runs until 2016, did not attend his pre-match press conference on Friday morning, ahead of the Premier League trip to Liverpool, with assistant David Kerslake taking the stage.
Journalists in attendance were also told not to tweet live from the press conference by Cardiff City.
Mackay has quickly been made odds on favourite by bookmakers to be the next Premier League manager to lose his job, while Ole Gunnar Solksjaer was tipped as his replacement.
Mackay, who brought Cardiff up to the Premier League as champions, has guided his side to 15th place in the Premier League after Saturday's 1-0 win over West Bromwich Albion.
In an interview broadcast on Tuesday on BBC Radio Wales, Mackay said: "I will not resign from my post. I was surprised obviously, but at the same time my job doesn't change.
"It's a job only three-quarters done as far as I'm concerned - making Cardiff a sustainable Premier League team.
"I have a great group of players who are competing at this level and enjoying this level. The fans are enjoying being at this level and my duty is to make sure we continue to do that.
"It would be wrong of me to walk away from a job because the owner doesn't want to give me more money in January."
On Thursday, Mackay received support from Liverpool manager Brendan Rodgers who Cardiff face on Saturday.
"I find it astonishing, what he's had to go through," said Rodgers, who worked with Mackay when he managed Championship side Watford in the 2008-2009 season.
"He's going to go on and become a big manager at a top club and I find it astonishing there's talk about him leaving there, absolutely astonishing.
"My only conclusion when I look from the outside is you've got a business guy operating the club who obviously knows nothing about football.
"He's (Tan) obviously been a very successful business man in his life, but football is like no other business."