The Slovenian played down the chances of him taking on the vacant post following 24 hours of intense speculation linking him to Leeds.
Milanic, 46, has emerged as the leading contender for the job at Elland Road after holding talks with Cellino last week.
The two men are understood to have discussed the role on Sunday, despite Sturm Graz claiming that neither they nor Milanic had received any contact from Leeds.
Milanic is under contract in Graz until the end of the 2015-16 season and he has been in his current position for little over a year.
Sources in Austria, however, believe Sturm would be willing to consider an approach for him after a mixed spell with Milanic in charge.
The club finished fifth in the Austrian Bundesliga last season and are in a mid-table position after eight games of the new term.
But both they and Milanic attempted to quell talk about his future, with Milanic saying: “In football there are always rumours but to me there is no offer from another club.
“I’m focused 100 per cent on the job with Sturm.”
A club statement added: “To bring clarity to the matter, there (has been) no contact between the clubs. Also there has been no contact between Milanic and Leeds.”
Milanic was singled out by Cellino as a possible option to replace David Hockaday last month and he came into the reckoning after United’s president failed to reach a deal with ex-Brighton boss Oscar Garcia.
Milanic – an ex-Yugoslavia and Slovenia international – moved to Austria in the summer of 2013 after a successful five-year spell in charge of Slovenian side Maribor, a period in which he won nine domestic trophies.
Cellino is believed to see Milanic, who grew up in the former Yugoslavia but speaks good English, as an ideal candidate to manage United’s cosmopolitan squad. Milanic’s relatively young age has also given him an advantage over more senior alternatives.
Alex Sabella, the man who led Argentina into this year’s World Cup and played for Leeds in the 1980s, is believed to have expressed a firm interest in the job at Elland Road but the 59-year-old is unlikely to come into the reckoning unless Cellino’s bid to land Milanic fails.
In the meantime, caretaker Neil Redfearn is preparing for a fourth game in charge at home to Huddersfield Town on Saturday with Leeds running short of time to appoint Hockaday’s replacement before the Yorkshire derby.
United have been without a head coach since Cellino sacked Hockaday on August 28 after just six competitive games in charge.