The Hammers are eager to add a new centre-back before the transfer window closes.
And Serbian international Maksimovic is high on their list with an offer made in recent hours.
West Ham are now in talks with Napoli to try and agree terms for the 28-year-old.
But Fulham seem to have prioritised Barcelona defender Jean-Clair Todibo, which puts West Ham in the driver’s seat for Maksimovic.
Maksimovic has been with Napoli since 2016, although spent the first half of 2018 on loan at Spartak Moscow.
The 6’4″ tall central defender came off the bench at half-time in Napoli’s 6-0 thrashing of Genoa at the weekend.