The Bluebirds confirmed on Friday that Iain Moody, one of Mackay's most trusted aides since their time together at Watford, had left his post permanently, having been placed on gardening leave earlier this week.
They also confirmed 23-year-old Kazakhstani Alisher Apsalyamov as Moody's replacement on an interim basis.
Apsalyamov is thought to be a friend of Tan's son and appears to have no previous experience in football.
The loss of Moody had placed Mackay's position under scrutiny but Press Association Sport understands the Scot has no intention of resigning, and the club have denied reports Mackay was asked to leave his post.
Tan said in a club statement: "I have every faith in Malky and his team to lead us through the challenges of the Premier League. I have supported him in the past and will do so in the future for many years to come.
"I would say to all Cardiff City fans and everyone connected to the club, let us look forward to the future and remain united in our support of the team."
The statement went on to thank Moody for his work for the club. Mackay has publicly praised his former colleague's diligence in helping bring the likes of Steven Caulker, Gary Medel and Fraizer Campbell to the club.
"Cardiff City football club can announce the departure of head of recruitment, Iain Moody," the statement read. "Iain is leaving the club with our thanks for his contribution.
"In the interim period, Alisher Apsalyamov has been appointed as acting head of recruitment, focusing on gathering data on individual players. Ultimate recruitment decisions of course remain the domain of the manager and majority shareholder."
Source: PA
Source: PA