The Belgian international has less than two years remaining on his current deal and talks between his representative and the Toffees are believed to be ongoing.
Moyes believes that the 23-year-old, who become Everton's record signing in 2008, will not look for new pastures like fellow midfielder Mikel Arteta did during the summer.
"I think it is very close to getting done," the Scot told the club's official website.
"The chairman and the chief exec have been working on it for a while and I've been waiting on the good news and hopefully it's not too far away.
"He has been a really good player for us over the last few seasons and if we can keep him it will be really good."
Fellaini has been an ever-present in Everton's starting lineup this season since recovering from ankle surgery.