Mackay's position looked to be under serious threat after it was reported earlier this week that Tan had told the Scot to resign from his post or be sacked.
Former Watford manager Mackay insisted that he would not stand down following Cardiff's 3-1 defeat at Liverpool on Saturday, with the travelling fans at Anfield vocalising their support for the 41-year-old.
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"I have spoken to Vincent Tan and he has agreed I can go into bat to try to bring a solution to this situation," Dalman said in a statement on the club's official website.
"The important thing is that we try to find a way through this predicament.
"There's goodwill on Vincent’s side. I've not had a chance to talk to Malky just yet, but I will obviously do that as well. We need to have further dialogue to see if we can get through this together for the good of the club, which is the most important thing.
"If we want to find a way forward, we now have a platform, a base, from which to work. In order to do that, it’s best we have an attempt at dialogue.
"I can tell you categorically that I, as chairman, have not initiated any dialogue with anybody else. I've read and heard so much nonsense about us speaking to four different candidates and that sort of thing. It's simply not true.
"If we don't succeed with our dialogue with Malky, I and the board will be handed the responsibility of going out and getting the manager we think is best to take the club forward.
"As things stand Malky is in charge for the foreseeable future and will be until something else happens."