The 24-year-old Ukraine international has been the subject of interest from a host of Premier League clubs over the past year.
Konoplyanka is thought to have a £12m buy-out clause in his contract and it is likely the Hammers will need to meet that clause in order to progress with the transfer.
And Sky sources understand they could face competition from at least one other Premier League club.
Konoplyanka was close to a switch to Liverpool in the January transfer window, only for the deal to break down on deadline day.
Tottenham have also previously expressed interest in the player but it is understood that neither they nor Liverpool have followed up their initial contact in recent weeks.
West Ham owners David Sullivan and David Gold have been active so far in the transfer market after setting manager Sam Allardyce a target of a top-10 finish in the Premier League.
They have signed Ecuador’s World Cup star Enner Valencia, forward Mauro Zarate, midfielder Cheikhou Kouyate, defender Aaron Cresswell and promising youngster Diego Poyet.
But Konoplyanka, with 28 goals in over 100 appearances for Dnipro and eight international goals for Ukraine, would mark a real coup for the club.