El Mundo Deportivo says the Iceland international has a firm offer from United on the table and also interest from Premiership rivals, including West Ham United.
Fenerbahce, which have already swooped for Real Madrid star Roberto Carlos, have a delegation, led by sports director Ali Gultiken, in Spain today to meet with Gudjohnsen about a move to Turkey.
Gudjohnsen doesn't want to a make a decision over his future until the end of the Spanish Liga season.