SkyBet Championship: Leeds United set to name new boss
The 46-year-old former Yugoslavia and Slovenia international is resolving his severance package from the Austrian club, reports BBC Radio Leeds.
Milanic won nine titles as a coach with his former club Maribor.
"Darko Milanic has received an offer from Leeds," read a statement published on Sturm Graz's Facebook page.
"Sturm have also been presented with a request from England. We will say more tomorrow."
Leeds are 11th in the Championship after Saturday's 3-0 win against Huddersfield, with Neil Redfearn as caretaker boss following the sacking of David Hockaday last month.
Milanic will be a new name to most Leeds fans having spent most of his playing career at Sturm Graz before he retired at 32. He has since managed in the Slovenian League with Primorje, Gorica and Mirabor before joining Sturm Graz in 2013.
Hockaday, whose only previous managerial experience had been with non-league side Forest Green, was appointed in June, the first manager recruited by controversial new owner Massimo Cellino.
Hockaday lasted only six games before he was sacked by Cellino, who dismissed 36 managers in 22 years in charge of Italian side Cagliari.
Cellino completed his takeover of Leeds in April and sold Cagliari the following month.
Brian McDermott, who was sacked in January when Cellino was poised to buy Leeds only to be reinstated a day later, left the club by mutual consent in May.