The move came just hours after Hammers co-owner David Sullivan insisted the entire first-team squad at Upton Park, excluding Scott Parker, were up for sale following a season of struggle in the Premier League.
Highly-rated Dorrans, 22, scored 17 goals to help the Baggies win promotion back to the Premier League and was voted the second best player outside the top-flight at the Football League Awards.
West Brom chairman Jeremy Peace insists the Scotland international is not for sale but the Hammers could yet come back with an improved offer.
Just days after securing their Premier League survival, Sullivan revealed every player, bar Parker, were for sale at the right price, while question marks also remain over the future of boss Gianfranco Zola, who is reportedly considering taking time out from the game.
l Meanwhile, Liverpool are set to sign Charlton’s highly-rated 18-year-old midfielder Jonjo Shelvey for an initial fee of around £1.7m.
Romford-born Shelvey held talks with Anfield chiefs yesterday and underwent a medical last night.