The former Sunderland defender has been without a club since being released by Lyon at the end of last season after being limited to just thirteen appearances in four years at the French club.
Mensah has been on trial at the Hammers alongside former Arsenal and Manchester United defender Mikael Silvestre and ex-Pompey man Anthony Vanden Borre. Allardyce has decided to not offer Silvestre and Vanden Borre contracts at the club after they failed to prove to the West Ham boss that they are good enough to compete in the Premier League.
The 35-year-old Silvestre may now well decide to hang up his boots on what has been a successful career winning five Premier League titles, a Champions League title as well as FA Cup and League Cup titles.
Allardyce bolstered his attacking options in the summer with the signings of Andy Carroll, Matt Jarvis and Yossi Benayoun. However, the Premier League newcomers still appear a bit short at the back with only six first-team defenders at Allardyce’s disposal.