Roberts enjoyed periods with both Oxford United and Shrewsbury Town last season, and the 18-year-old has been given permission to spend a similar spell at near neighbours Walsall.
Whitney told the club's official website: "He's been one of our key targets through the whole of the summer but because he's done so well, West Brom have kept him for their first team.
"It's nice to get it over the line and they see him as a really good fit for us. Bringing Shaun in and playing him has helped us, because they see a pathway and important match time.
"He'll be a very important member of our squad and he's a tremendous prospect. It's great for us to bat off a number of clubs in trying to get his signature, and for West Brom and the player to put their trust in us."
During his time in League One last season, Roberts scored a total of four goals in 11 starts and 16 substitute appearances.