The Wales international defender completed a medical and agreed personal terms at Albion’s training ground on Tuesday ahead of a move which Sky Sports sources believe is worth £8m.
The 26-year-old, a former Manchester United player, made 171 appearances for Hull and scored eight times, including his side’s opener in the 3-2 FA Cup final defeat to Arsenal in 2014.
Head coach Tony Pulis told West Brom’s official website that a glowing reference from former boss Steve Bruce convinced him to sign Chester.
He said: “James did very well for Hull City. I’ve had a good chat with him and he’s the type of player I like.
“He’s young, a strong character, a good defender, he has good feet and he has the Manchester United stamp all over him, just like (Stoke City’s) Ryan Shawcross and Darren Fletcher. They’re wonderful lads and I’m told James is as well.
“From the people I spoke to, like Steve Bruce and people in the Welsh party… they speak very highly of him.
“Steve especially gave him a great mention and there is no better accolade than Steve Bruce giving a defender a good drum roll.
“James is a good player who can play anywhere along the back line and I’m confident he will get better with age.”
On transfers in general, Pulis said West Brom would only bring in players if they were what the team needed.
He added: “We’re trying to bring in others, but they have to be the right players. We’re very pleased to have James on board and hopefully we can work hard to get more in.”