Chester, 26, is due to have a medical at The Hawthorns on Tuesday afternoon.
The former Manchester United defender has one year left on his existing deal at the KC Stadium and the Tigers risk losing him for nothing at the end of the season.
He has made 171 appearances for Hull – scoring eight goals – after signing from Old Trafford in 2011 and is set to become part of Tony Pulis’ squad at West Brom.
Pulis has signed Wigan winger James McClean this summer and Sky sources understand the Baggies boss has also held talks with Liverpool striker Rickie Lambert.
Former Arsenal midfielder Abou Diaby had a medical with the club but he has rejected the chance to move to West Brom and instead is looking for a club in France, according to Sky sources.