The Senegalese striker had looked to be on his way to Craven Cottage, but with the clubs still haggling over a fee and Heidar Helguson stalling over a move in the opposite direction, the Hammers have taken advantage by tabling a £5.5million offer, according to the Daily Mirror.
Kamara could seal his move as early as Monday after indicating a willingness to move to Upton Park and fight for his place.
Hammers boss Alan Curbishley wants plenty of options in attack, with Craig Bellamy close to completing his move from Liverpool, while Carlos Tevez and Marlon Harewood are both on their way out of the club.
Kamara, who scored 23 goals in the Championship last season, moved to England in 2004 when he signed for Portsmouth for a then-club record fee of £2.25million from Italian side Modena.
Despite failing to settle at Fratton Park, the Baggies snapped him up for £1.5million and he has become a fans favourite at the Hawthorns, attracting the attentions of a host of clubs in the Premier League and in Italy.