Mackay's position is under intense scrutiny after it emerged that Tan had sent him an email threatening to fire him unless he quit.
It is also believed Cardiff have already approached four potential candidates to replace Mackay as Bluebirds boss.
However, it appears the 41-year-old Scot, who has travelled to the north-west with his squad ahead of Saturday lunchtime's Premier League game at Liverpool, will not bow to Tan's demand.
Tan arrived in the UK by private jet on Friday and met chairman Mehmet Dalman in London to discuss Mackay's future before heading to Merseyside.
Tensions have been running high between Tan and Mackay, with the owner accusing his manager of unfairly raising the expectations of the club's supporters by suggesting he was looking to sign three players in the January transfer window.
Mackay was absent from the club's Friday press conference after it emerged that Tan had contacted the Scot on Monday, outlining his grievances at length and issuing his ultimatum.
Malaysian businessman Tan was allegedly critical of Mackay's signings, style of play, results and overall performance as a manager.
He is also believed to be unhappy at an overspend on players during the summer, with £15m more being spent than the originally slated budget of £35m.
In Mackay's absence, assistant boss David Kerslake told reporters at the conference that the manager was still in charge, adding: "No one has said any different to me, everything has been as normal. Players have got on with their jobs.
"Without being boring, it has been a normal week."
When Kerslake came in for sterner questioning on Mackay's position, a club spokesman stepped in to say: "Malky Mackay is unavailable to speak to the media, the assistant manager is here."
Mackay insisted earlier this week that he would not walk away following a fresh rebuke from Tan, who said he would not receive 'a single penny' of transfer funds in January.
Mackay said: "I will not resign from my post. I was surprised obviously, but at the same time my job doesn't change.
"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."
There have been doubts about Mackay's future with the Welsh club since Tan removed his head of recruitment Iain Moody in October amid suggestions of a summer transfer window overspend, an accusation denied by the coaching duo.
Mackay took Cardiff into the top flight at the end of last season for the first time since 1962.
They are currently 15th in the Premier League table, four points clear of the relegation zone.