The Hammers have experienced a disappointing summer transfer window and they have yet to make any new additions to the first-team squad.
The east London outfit thereafter looked at the possibility of signing Wesley Fofana from Saint-Etienne, but they have missed out on his services to Leicester City.
The club are still on the search for a new player to bolster the heart of their backline and it is now reported that they have made a formal bid for Milenkovic.
David Moyes’ have placed a £22.8m (€25m) offer on the table for the Serbian, but it is suggested that Fiorentina are holding out for a higher package.
Milenkovic has a dominating aerial presence with his height of 1.95m and he has caught the eye with his good positioning in the Serie A.
Aside from his regular central defensive role, he is also capable of playing in the right-back spot, where the Hammers have just one senior player in Ryan Fredericks.
The Hammers were willing to pay £30m for both Tarkowski and Fofana earlier this month and they should be able to come up with an improved offer for Milenkovic.
Stats from Transfermarkt.com