The Hammers have rejected bids from Fulham, Birmingham and Stoke for Cole.
Cole joined West Ham from Chelsea for £1million in the summer of 2006 and the striker has become a key player at the club.
Reports suggest Arsenal and Aston Villa are also interested in the striker.
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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
Who of Europe’s top strikers will move to England this summer?