The 24-year-old Ukraine international has attracted interest from a number of Premier League clubs, but the Hammers are hoping to win the race for his signature.
However, Konoplyanka is thought to have a £12m buy-out clause in his contract and the east London club may need to meet that clause in order to progress with the transfer.
Liverpool made a bid for Konoplyanka in the January transfer window but failed to agree a deal before the deadline, while Tottenham and Manchester United have previously been linked with the player.
West Ham owners David Sullivan and David Gold have been active in the transfer market this summer after setting manager Sam Allardyce a target of a top-10 finish in the the Premier League.
Forward Mauro Zarate, midfielder Cheikhou Kouyate and defender Aaron Cresswell have already arrived at Upton Park, but the signing of Konoplyanka would be a real coup for the club.
Konoplyanka has made over 100 appearances for Dnipro since his debut in 2007, scoring 28 goals, and he has 36 caps and eight goals for Ukraine.