West Ham co-chairman David Gold confirmed the player had passed the medical ahead of a £15 million move.
Carroll spent last season on loan from Liverpool at Upton Park but had delayed making a decision on whether to move permanently.
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Who of Europe’s top strikers will move to England this summer?
After a dreadful start to their Bundesliga campaign, Borussia Dortmund appear to have weathered the storm after going five matches without defeat.