The veteran England international striker has accepted a one-year contract offer from the Hammers, says the Athletic.
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Wilson will join West Ham after turning down a pay-as-you-play contract offer from Newcastle after his previous deal expired in June.
Wilson has battled with multiple injuries at Newcastle over the past two seasons.
The 33 year-old has passed his medical at West Ham and will soon ink a 12-month contract which is heavily incentive-based.