The Sun says the Hammers want to lure the 25-year-old away from Norwich City in January.
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Manager Nuno Espirito Santo sees the striker as a potential replacement for Niclas Füllkrug (32), who could possibly return to the Bundesliga.
Sargent joined the Yellow Canaries from Werder Bremen in the summer of 2021. Since then, the US international has scored 55 goals and provided 15 assists in 154 competitive matches for the English Championship side, who are currently second-to-last in the standings and battling relegation.
West Ham hope to do business for around £15m for Sargent.