The Hammers have agreed a fee of €30m with CSKA Moscow for the transfer of attacking midfielder Vlasic, who has scored 33 goals and supplied 21 assists since joining the Russian club from Everton in 2018.
A deal to bring the 23-year-old back to the Premier League is now dependent on personal terms being agreed and a medical being completed.
Moyes has been looking to add to his ranks in the attacking areas this summer having seen Jesse Lingard return to Manchester United after his loan spell last season.
There has been interest in bringing Lingard back to London Stadium but the England international has been adamant that he wants to remain at Old Trafford and impress manager Ole Gunnar Solskjaer.
The arrival of Cristiano Ronaldo at United may change that but any return to West Ham seems unlikely should Vlasic’s move go through.
A striker has been a top priority for Moyes, though the arrival of Vlasic may see the Scot accept that he has enough options in the forward line, with Andriy Yarmolenko and Jarrod Bowen both able to play through the middle.
Vlasic will provide continued competition for Pablo Fornals and Said Benrahma, who have both made bright starts to the season.