Maksimovic is currently a free agent, having left Napoli last week when his contract expired.
West Ham, who are keen on bolstering their squad ahead of a season in which the Europa League is also on the agenda, have zeroed in on the Serbian as a top target.
And it has been claimed the Hammers have tabled Maksimovic an offer in the hope of adding him to their ranks for free in the ongoing window.
However, West Ham could lose out on a move for Maksimovic, as according to Italian daily Corriere dello Sport, he is yet to completely draw the curtain on his Napoli career.
The 29-year-old has not ruled out the possibility of signing a new deal at the Stadio Diego Armando Maradona despite interest from elsewhere.
It is suggested Napoli are also warm to idea of Maksimovic returning to the club on fresh terms, having seen him turn down a contract extension prior to his exit.
In addition to West Ham, La Liga outfit Real Betis are also keen on a move for Maksimovic, but he could remain in Naples for another stint.
