The 26-year-old was hospitalised last week, with the club initially confirming in a statement that Garcia was suffering with a "blood disorder".
However, Mackay told a news conference on Monday that the Honduras international had been diagnosed with a form of leukaemia.
"We've had some bad news as far as that's concerned – he has a form of leukaemia," the Scot told reporters.
"I went in to see Juan Carlos and he was incredibly upbeat and his spirits were strong.
"He's going to fight this and make sure he gets through this for the sake of his wife and kids. Everybody at the club is wishing him all the best as I'm sure that all our fans are.
"It's something that is obviously a real blow for Juan Carlos' family and everybody connected with the football club. We're going to make sure that he gets the best care possible."
