West Ham joint-chairman David Sullivan has told supporters a fifth signing can be expected in the next 72 hours.
A number of Championship clubs are keeping tabs on Leyton Orient defender Mathieu Baudry, Sky Sports understands.
The 21-year-old, who played in a 5-1 Capital One Cup defeat at St Andrew's in August, impressed City staff during a recent trial period with the club.
Manager Lee Clark told the club website: "We have been scouting Olly for a while now and he was one of Barnet's standout performers during our tie earlier this season.
"Olly has impressed us further during a recent trial period and we now look forward to seeing how his development continues over the coming months."
Lee, son of former England international Rob Lee, began his career as a West Ham trainee before signing for Barnet last summer.