Moyes' current contract at Goodison Park expires in the summer and he insists no discussions will be held over a possible extension until the end of the season.
Everton are currently sixth in the table, four points adrift of Arsenal in fourth following their 0-0 draw at Emirates Stadium on Wednesday, and look to be outsiders to force their way into the continent's elite competition.
But Belgium international Fellaini feels Moyes deserves huge credit for helping the Merseyside outfit challenge the established order despite operating on a transfer budget which is dwarfed by the spending power of the Premier League's leading lights.
Fellaini told reporters: "In 10 years he has made the club stronger and has positioned us just outside the top four or five clubs.
"We don't have the same budget as Manchester United, Arsenal, Chelsea or Tottenham either, yet we are still doing well.
"Playing in the Champions League would be great. We have a good team and I don't think we are far from it. But we have to work hard and win our games."