The Belgian international has made a superb start to the season, forcing Moyes to admit that he may leave the club in January.
Fellaini told L'Equipe that Moyes has helped him develop during his time at Goodison Park.
He said: "He has an amazing eye. Before signing me, he came several times to see me in my league in Belgium.
"With him, you work very hard. He is very demanding, but very close to his players too. Everton is just like a family."
The 24-year-old has three Premier League goals to his name so far this season, including a winner against Manchester United on the opening day.