Mackay is reported to have reacted furiously to the news his head of recruitment Iain Moody was being replaced by a 23-year-old family friend of the club's Malaysian owners.
Moody, who previously worked with Mackay at Watford, was suspended from his role to allow Kazakh Alisher Apsalyamov to step in on an acting capacity.
Moody and Mackay had added a number of high-profile signings to the squad this summer including England defender Steven Caulker and Chilean midfielder Gary Medel, as Cardiff returned to the top flight for the first time in 51 years.
Speculation immediately surrounded the future of Mackay, who joined City in the summer of 2011, but Cardiff are playing down reports of a rift despite most bookmakers cutting him to 6/4 (from around 25/1) to be the next Premier League manager to lose his job.
Sky Sports understands that Mackay has 'not been asked to resign in the row over staff changes' and that the Scot has 'the full backing of the dressing room and has no intention of quitting'.
Reacting to the unrest at the club, Tim Hartley, chairman of the Cardiff City Supporters' Trust, said: "Fans will be concerned with the recent behind-the-scene developments at Cardiff City which appear to put into question the future of our manager, Malky Mackay.
"Supporters are surprised that who appears to have little hands-on experience has been put in charge of transfers at the football club.
"Fans want Malky Mackay, who has done a fantastic job as manager taking the club to the Premier League, to remain at Cardiff City.
"We don't want to see any action taken to undermine his position as manager and Cardiff City's efforts to have a successful first season in the top division.
"There are fears that the current developments could drive Malky into the arms of another Premier League club and that would be hugely damaging for the Bluebirds."