In a bid to improve his defence, Rodgers is believed to be interested in a deal to sign Hammers centre back Reid in a cut-price deal.
The New Zealand international, 26, is out of contract at Upton Park this summer, and doesn't appear to be signing a new contract to stay in East London.
So far, the Hammers haven't had to consider any bids for his signature, but Liverpool could now test their resolve before Monday's deadline.
The Mirror report that West Ham value Reid between £6m and £7m, with Liverpool prepared to pay £6.5m to take him to Anfield.
Rodgers has seen summer signing Dejan Lovren struggle since his move from Southampton, leaving Emre Can to play in an unfamiliar centre back role.
Adding Reid would certainly help matters, and if they can get him on the cheap, it could be superb business by Rodgers and Liverpool.
Reid is able to leave for free this summer, and it may be wise for West Ham to take the money whilst they still can - and finally put the saga to bed.