The 20-year-old had been expected to link up with the Baggies after manager Tony Mowbray revealed his interest over the weekend.
Both clubs have now confirmed that Simpson will spend the rest of the season at the Hawthorns to build up his first-team experience.
However, he has yet to make a Premier League appearance for Arsenal and Mowbray recently admitted he is taking a gamble on the youngster.
"I would suggest Jay is an unknown quantity at Premier League level," conceded Mowbray. "Can he score goals? I don't know.
"But we'll hopefully give him an opportunity to see if he can.
"He's obviously had a very good upbringing at Arsenal and every player needs an opportunity."