Simon Lim claimed on Monday that owner Vincent Tan would not make a "single penny" available for transfers during January after concerns of over spending during the summer
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, adding that he felt the club has overspent during the summer.
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"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, who lie 15th in the Premier League table after winning just two of their last 10 games, broke their transfer record three times during the summer with the purchases of Andreas Cornelius, Steven Caulker and Gary Medel.