The 32-year-old striker, available on a free transfer after leaving Stoke, was poised to join the Hammers on a one-year deal after passing a medical at Upton Park.
But boss Sam Allardyce, who worked with the Iceland international during his time in charge of Bolton, has decided to end his interest after Gudjohnsen hesitated in putting pen to paper, with reports in Greece suggesting AEK Athens have instead offered him a two-year contract.
Change of heart: Gudjohnsen had a medical at West Ham
New West Ham manager Sam Allardyce wanted to add the Iceland international to an attack which has been shorn of Demba Ba, Robbie Keane and Victor Obinna following the club's relegation, but will now look elsewhere.
Former Chelsea and Barcelona forward Gudjohnsen came to prominence under Allardyce at Bolton, where he scored 27 goals in 77 appearances before making a ?5million switch to Stamford Bridge in 2000.
History lesson: Eidur Gudjohnsen (right) played under Sam Allardyce at Bolton
Gudjohnsen has come close to moving to Upton Park before, in January 2010, only to change his mind and join Tottenham instead.
The indication of AEK's interest comes less than 24 hours after Hammers striker Carlton Cole turned down a ?6million transfer to Stoke.
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