Chief executive Simon Lim claimed Tan was "extremely upset" by Mackay's public revelation that he wanted to sign three new players in the January transfer window.
Lim's statement on Monday also revealed that Tan felt Mackay had overspent in the close-season, with the likes of Gary Medel, Steven Caulker and Andreas Cornelius recruited for big-money fees.
As a result, Lim said Tan would not make a "single penny" available for Mackay to improve the playing squad when the transfer market opens in January.
But the Scot says he is unperturbed by the claims, and believes it would be "wrong" to walk away from the Welsh outfit.
"I will not resign from my post," Mackay told BBC Radio Wales. "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 am 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 money in January.
"I have been in situations before when a board has had no money."
Cardiff lie 15th in the Premier League table and travel to high-flying Liverpool on Saturday.
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