Reds boss Roy Hodgson is thought to be interested in taking Cole to Anfield, especially after the former Fulham boss failed in his approaches for PSV's Ola Toivonen and Bayern Munich's Mario Gomez.
Hodgson has spoken of his desire to add at least one more striker to his squad to take the pressure off star man Fernando Torres this season.
And despite the Hammers' desire not to sell their leading hitman, Cole would welcome a move away from east London to one of the Premier League's top clubs.
Avram Grant is keen to keep Cole, 26, at Upton Park, but it is thought the club's owners would be tempted to cash in at the right price.
In other news from the club's training ground at Chadwell Heath, much-discussed midfielder Scott Parker arrived for this morning's fitness session.
However, speculation continues to surround his future - especially after the player spent several minutes on his mobile phone while waiting in the car park.
Rumours are currentl circulating he has been in contact with his agent numerous times over the past few days as offers of big-money moves have come from several English clubs.
