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 – who guided Cardiff to promotion from the Championship last season - insisted that he would not stand down following their 3-1 defeat at Liverpool on Saturday, with the travelling fans at Anfield vocalising their support for the 41-year-old.
Chairman Mehmet Dalman confirmed on Sunday that Mackay will remain in charge for the "foreseeable future", but added that the club will seek a replacement if he cannot open a successful dialogue with the club's controversial Malaysian owner.
"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."
The loss at Liverpool left Cardiff only four points ahead of the relegation zone in 15th place, and Dalman is keen for the situation at the club to be stabilised to aid their fight for Premier League survival.
"If we want to find a way forward, we now have a platform, a base, from which to work," he added.
"This is a crucial time in the Premier League. Every point counts and we are better off going for stability. So if we want to give the club the best chance of being in the Premier League next season, the best way is to fix the problem and seek stability.
"In order to do that, it's best we have an attempt at dialogue. Vincent Tan is in London for a while, so at least he is physically here which makes it easier to have that face to face 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."
Dalman also called on fans to end their protests against Tan and instead preach a message of unity in their upcoming games.
He continued: "If I had one wish for Boxing Day, it's that they (the fans) don't hold up 'Tan Out' or 'Malky In' banners, but rather one that says 'Talk to each other'.
"The crisis for the time being is over. The emphasis as of today is for us to create space and dialogue."
