The Yorkshire outfit are yet to replace Dave Hockaday - sacked in August after delivering only one Sky Bet Championship victory during his 70-day tenure - but Slovenian tactician Milanic has been asked to take the job.
Shortly before Sturm Graz beat SV Ried 1-0 in the Austrian Bundesliga on Saturday, a statement published on their official Facebook page read: “Darko Milanic has received an offer from Leeds.
“Sturm have also been presented with a request from England. But now we concentrate fully on the game. We will say more tomorrow.”
Should Milanic, 46, fly to Leeds in the coming days and subsequently agree to work under the club’s ambitious Italian owner Massimo Cellino, his first game in charge would be next Saturday’s trip to Brentford.
The 11th-placed Whites are unbeaten under the supervision of caretaker manager Neil Redfearn, who oversaw a 3-0 home victory over Huddersfield on Saturday and who is likely to be closely consulted by Milanic on first-team team matters during the early weeks.
As a player, Milanic captained Slovenia at Euro 2000 and, upon turning to management, guided Maribor to four Slovenian league titles and three Slovenian Cup wins.
He joined Sturm Graz on a three-year contract in June 2013, guiding the Styrian club to fifth-placed finish in 2013-14.
Die Schwoazn currently lie seventh in the Bundesliga table.
Speaking during the match against Ried, Sturm general manager Gerhard Goldbrich told diepresse.com: “Milanic has a tremendous opportunity in England. We do not want to punish him.”