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The Icelandic forward, also a target for Reading, has failed to impress boss Tony Pulis since his summer transfer from Monaco, and has only made five appearances for the club, all from the bench, without scoring.
Pulis is said to be unhappy at the former Chelsea and Barcelona star's overall fitness levels and would be willing to listen to offers for the striker.
Stoke are currently sitting in the top half of the Premier League and Pulis may feel that he has adequate cover to cope with the possible departure of Gudjohnson.
However, Fulham may have competition for his signature with Swansea City manager Brendan Rodgers said to be contemplating a move for the player, according to The Daily Mail, having previously worked with Gudjohnsen during their time together at Chelsea.