The former Sunderland star is available on a free transfer after ending his contract with Lyon during the summer.
Mensah impressed last season while on loan at Sunderland.
The 29-year-old is a free agent and is able to hold talks with any club despite the transfer window closing on Friday night.
The Daily Star reports that the Hammers are keen to agree a deal with the defender as soon as possible.
What is your opinion? Feel free to discuss the article under. You have to comment by using your Facebook profile. Be fair and have respect for other people and opinions!
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.