The 24-year-old Ukraine international is expected to leave his homeland in the coming months after the Reds came lose to securing his signature in January, only to see their efforts thwarted as the transfer deadline passed.
Since then their interest is thought to have cooled, which has left the door ajar for the Hammers to launch an audacious swoop.
The Londoners are understood to have tabled a £12m bid, which meets his reported release clause.
However, there are indications that the winger would be reluctant to relocate to Upton Park, even for a big salary.
‘Kono’ is known to want Champions League football, which he could achieve by staying with Dnipro.
West Ham are still hopeful that they can nab the Eastern European, who could slot into their first-team as Sam Allardyce plots to employ a more attractive playing style next term.
Manchester United are also keen and Liverpool may up their interest once again after agreeing personal terms with the player over the winter.