Old Trafford bosses have launched an enquiry to discover the Serbia international’s availability.
A deal for £90m-rated Leicester City ace Harry Maguire is proving difficult to agree so Ole Gunnar Solskjaer is scouring for alternatives.
Manchester United considered a swoop for Milenkovic in January.
According to Gianluca Di Marzio, Fiorentina knocked back a £35m (€40m) offer from the Red Devils in January.
The Serie A outfit valued Milenkovic at £44m (€50m) at the time so are likely to ask for a similar price.
Solskjaer wants a commanding presence to partner Victor Lindelof at the back next season.
England international Maguire is the preferred target but Milenkovic would be a cheaper alternative.
And fans have urged transfer kingpin Woodward to go and strike a deal for the 21-year-old.
“Milenkovic would be a great acquisition. This is one deal that I would defo love to see happen, but we all know Ed and Matt Judge will mess it up somehow,” one supporter wrote on Twitter.
“Milenkovic has way more potential than maguire,” a second commented.
“Milenkovic > Maguire,” a third added.
Another said: “Let me make this clear: Milenkovic is not a perfect player. He is not a saviour for our backline. He makes mistakes. But I love him. Never shy of making a tackle. Would love to see him at Old Trafford.”
While a fifth added: “I fully expect this to be smoke and still buy Maguire for 80m. And for the record I’d buy Milenkovic at €50m before I buy Maguire at £80m.”
United have completed two signings so far this summer.
Daniel James was first through the door from Swansea City before a £50m deal was agreed for Aaron Wan-Bissaka.