Several of Tuesday morning's papers have linked the Icelandic ace with a move to Upton Park this summer as the Hammers continue their search for new faces.
The arrival of Thierry Henry at Barcelona has left Gudjohnsen well down the pecking order and he is now highly expected to make a return to England, where he last played for Chelsea.
West Ham's approach for the 28-year-old forward has been held up by the ongoing transfer saga of Carlos Tevez but with his exit now in sight, chief executive Scott Duxbury can now turn his full attention to the club's new target.
According to the Daily Star, Gudjohnsen would cost around £7million but the Daily Mirror and Daily Express suggest the fee could be as much as £10million.
Gudjohsen's father and agent Arnor, said on Monday: "West Ham would be something Eidur would consider, as there are some Icelanders there with the owners.
"Barcelona have always told him it is up to him what he wants to do, and they will not stand in his way if he wants to go."