Stoke reportedly bid £3 million for the 27-year old but West Ham are holding out for a fee closer to £7 million for the England International.
Cole began his career at Chelsea before spending loan spells at Wolves, Charlton and Aston Villa. West Ham signed the player on a permanent basis in 2006 and he has since gone on to make 158 appearances for the Hammers, scoring 42 goals on the way.
His 11 goals last season were not enough to save the East London side from relegation and now the club could cash in on their star striker, who is reported to be one of the highest earners at Upton Park.
Stoke are also reportedly interested in Preston midfielder Adam Barton and are willing to offer £1 million for the 20-year old. The club have also approached Birmingham about a double deal for defender Scott Dann and striker Cameron Jerome but must increase their initial £16 million bid for the pair.