The Hammers are on something of a recruitment drive this summer and have already completed a number of ambitious new additions, including Dimitri Payet and Angelo Ogbonna.
Slaven Bilic’s men are clearly keen to further strengthen ahead of the 2015-16 campaign and look to have identified Mirallas as a player to bolster their ranks.
HLN state that the London club have bid €8.5 million (£6 million) for the Belgium international, but Roberto Martinez’s men have turned the offer down.
Mirallas has had an up-and-down time with the Merseyside club, with his future with the Toffees a point of contention on a regular basis.
The fact that the bid has been rejected would insinuate that Martinez sees the versatile forward as a player he needs in the forthcoming campaign.
Should West Ham come back with an improved offer and land the former Olympiakos attacker, their attacking options ahead of next season all of a sudden would be very comprehensive.