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.
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The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.
As always it is the so called “big clubs” that have splashed the most cash in the transfer window so far this summer. Manchester United or City? Arsenal or Tottenham? Who are the winners and losers of the transfer window so far?