Having joined the Shrimpers on a three month loan, the 19 year old was injured within the first hour of training.
Blues boss Phil Brown said: “Elliot came to us fully fit but 35 minutes in to his first session he pulled up with a muscle strain.n.
“He’s gone back to get that assessed at West Ham and will have a scan.
“That will tell us more but hopefully it won’t be too bad.”
Lee will be with Blues until January and has been given permission to play in the FA Cup when fit.