The former Slovenia international defender was in charge of Maribor in his homeland, where he won the Slovenian League four times, the Cup three times and the Supercup twice.
However he's struggled since moving to Sturm Graz in 2013 finishing fifth in his first season in charge and will leave with the club currently sixth in Austrian Bundesliga.
The 46-year-old former Slovenia and Yugoslavia international has a clause in his deal with Sturm Graz which allows him to buy out of his three-year deal.
Leeds are now understood to have activated that clause, though they have yet to confirm whether Milanic, who guided the Austrian side to a mid-table position last season, is the new manager.
If appointed he will replace David Hockaday, who was sacked after just six games in charge at Elland Road.
But will inherit a side rejuvenated under caretaker Neil Redfearn. Leeds have won three of their last four and now sit 11th in the Championship, just two points outside the playoffs.