The Welsh international was signed in May 2005 but he has failed to make any real impact on the team during his 35 games since his switch from Spurs.
He has scored just one goal during his time at Everton, that mad bobbling long range winner at Birmingham last season.
The fact that Moyes has let Davies talk to Fulham suggests that all is going well with the proposed move for Manuel Fernandes who had a medical at Bellefield yesterday.