Mackay will now have to work with 23-year-old Alisher Apsalyamov, who is from Kazakhstan and is a friend of Tan's son.
He has no football experience but had been helping out with jobs around the stadium.
Moody joined Watford as press officer while Mackay was manager before becoming the football operations manager at Vicarage Road.
Since following Mackay to Wales, Moody has played a pivotal role in the acquisitions of the star names such as Steven Caulker, Gary Medel, Kevin Theophile-Catherine and Fraizer Campbell.
The decision to suspend Moody comes just days after it was revealed Tan and the City squad had been at loggerheads over end-of-season bonus payments.
Cardiff are making no comment for legal reasons, but bookmakers Sky Bet have cut the odds on Mackay being the next manager to part company with his club in the wake of the news.
In an industry-wide gamble, Mackay was slashed from 40/1 to as short as 6/4 to be the next top-flight manager to part company with his club.
The decision to suspend Moody, whose work in the scouting and recruitment of players has been publicly praised by Mackay, comes just a matter of days after it was revealed Tan and the Bluebirds squad had been at loggerheads over end-of-season bonus payments.
Mackay confirmed following Saturday's 2-1 defeat to Newcastle that a resolution has been reached over the issue.
Sky Bet's Dale Tempest, whose firm cut Mackay's odds to 13/8, said: "While Cardiff haven't had the greatest start this season, of more concern to Mackay has to be that the owner appears to be making decisions without the manager's agreement.
"When that happens, it's no surprise to see money across the board."
Previous favourite, Fulham's Martin Jol, drifted out to a best price of 5/1 with Sky Bet as a result of the money for Mackay.