Former West Brom centre-back Dawson, who has made 246 Premier League appearances, spent the past two seasons at the Hammers.
Wolves, who are 17th in the Premier League, face a trip to Manchester City (14:00 GMT) on Sunday.
Under new manager Julen Lopetegui the club have signed midfielder Mario Lemina, winger Pablo Sarabia and forward Matheus Cunha.
Dawson, whose last appearance for the Hammers was in an FA Cup win at Brentford on 7 January, was not registered in time to face City.
He initially joined West Ham on loan from Watford in October 2020 before turning that into a permanent move.