Mackay confirmed on Monday that he expected to be relieved of his duties over the weekend, having received an email from Tan asking him to resign or be sacked.
Although Cardiff are seeking to end the row between their manager and owner, Dalman has warned the issue cannot be allowed to drag on as the club attempt to preserve Premier League status following last season's promotion.
Dalman told Sky Sports: "Every effort that can be made is being made to allow the club to move forward.
"We will do everything humanely possible to maintain the status quo, but if not we will move fast to repair the situation.
"If that means bringing in a new manager, we will - but for the time being, we believe that's premature and we will work together to fix the predicament we're in."
In his press conference on Monday, Mackay claimed his representative had been rebuffed when attempting to arrange a meeting with Tan.
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