The 22-year-old has agreed a five-year contract, with the Hammers describing the fee as a record for an EFL Championship club.
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The deal surpasses the £17.5m Ipswich Town paid for Sindre Walle Egeli last year. Following their relegation in May, West Ham begin their first Championship campaign since 2012 away to Burnley on Sunday.
Engels, who can operate in midfield, at right-back or on the right wing, will make his debut later.
He is unavailable for the Burnley trip but could feature against Charlton Athletic on August 22.
Engels won consecutive Scottish Premiership titles with Celtic after joining from Augsburg in 2024.