The 17-year-old has signed a one-year contract with the Baggies after leaving Arsenal by mutual consent earlier this summer.
West Brom’s sporting and technical director Dan Ashworth said: “Arsenal let us have a look at Jamie for a three-week period during which he played in two reserves games and a couple of youth team games.
“He impressed us with his ability on the ball, his discipline and willingness to improve.
“Therefore, he has earned himself a one-year contract and the chance to progress his career here.”