West Brom have sacked Nicolas Anelka citing "gross misconduct" after the striker claimed on Twitter he had quit the club.
Mauricio Pochettino expects to be at Southampton next season and is looking forward to working with chairman Ralph Krueger.
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.