The Scotland international, 29, was substituted in the seventh minute of Wednesday's 2-2 draw at Stamford Bridge and missed Saturday's 3-0 Premier League defeat by Southampton.
Morrison, who is out of contract in the summer, has made 19 appearances for the Baggies this season.
He is the club's joint top scorer this season with four goals.
West Brom midfielder James Morrison is facing up to two months out with the knee injury he picked up at Chelsea.