City are keen to keep the right-back as they are struggling to fulfill their homegrown quota for next season but Richards is sais to be unwilling to sign a new deal after playing a peripheral role under Manuel Pellegrini.
The expected arrival of Bacary Sagna will see Richards fall even further down the pecking order and the 25-year-old is now set for a new challenge.
A host of Premier League clubs are reportedly interested in signing Richards, including Arsenal and Liverpool, although a move to West Ham would guarantee him regular first-team action.
With only 12 months remaining on his contract, City would demand a transfer fee in excess of £5million but West Ham would surely be willing to pay that for a quality defender.
His reported £70,000-a-week wages could be an issue, but the lure of first-team action will help in any negotiations.
Allardyce used Ivorian Guy Demel at right-back for much of the last campaign though he can also fill in at centre-back, and alongside Richards would likely play in the heart of defence rather than at fullback.
The City man is more dangerous as a fullback and his power and pace make him a devastating threat going forwards as well as in defence.
Allardyce is often criticised for his defensive approach but Richards would tick all the boxes as he offers defensive stability as well as an outlet in attack, and with Demel now 32-years-old, he could breathe new life into the Hammers defence.
He has been at City for nine seasons but has only made nine appearances in the Premier League over the past two seasons, with only two in the last campaign.
A move to the Hammers would give the defender a new lease of life at the perfect time in his career and he could shine once again.
With so many question marks over England's right-back position, Richards could also force his way back into the picture with a move after two years without an appearance for the Three Lions.