Wilson, 34, signed a pay-by-play deal with West Ham last summer following his release from Newcastle and has since scored five goals across all competitions.
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The veteran striker came close to leaving in January but ultimately remained at the club as a squad player for Nuno Espirito Santo as they battle against relegation.
According to The Telegraph, Wilson is set to be rewarded with a new one-year contract by the East London club.
West Ham currently sit in 18th with 28 points from their 29 games, level with Nottingham Forest and just one off Tottenham.