Moyes will officially replace the retiring Sir Alex Ferguson at Manchester United on July 1 and Fellaini is sure his fellow Scot has all the tools to succeed at Old Trafford, including Fergie's famous 'hairdryer treatment'.
Fellaini was personally disciplined in a gruelling training session after headbutting Stoke defender Ryan Shawcross last season, as well as being fined two weeks' wages, and the Belgium international believes the lesson Moyes taught him that day proves the 50-year-old will not be fazed by the job ahead of him.
|20/1||Everton are 20/1 with bet365 to finish in the top four next season
"Football is like this. He can be scary.
"When I was banned after the Stoke game, he gave everyone the day off – the players, the entire staff. But I came to training to do a session with him.
"It was not easy to go into the camp to face him like that. I understood, because I knew I had made a mistake.
"But, you know, it's not easy when you know you have let him down. Just going in there knowing you have to look him in the eye was tough!
"He put me through a tough session. It was a full on: running, shooting, passing. Everything was very intense and afterwards I was upset.
"I said afterwards 'this is bad' but I am happy when he looked after me, so I understand."
Fellaini, whose own future is far from secure, thanked Moyes for giving him the chance to prove himself in England and believes the Scot should be applauded for constantly overachieving with an Everton side operating on a shoestring budget.
"He was so important to me, he gave me my chance and he has done everything for me," the Belgian added. "He gave me the confidence to play at this level, I learnt a lot from him.
"There's no doubt David Moyes is a top manager who can be at Manchester United. Every year with Everton we are in the top six, we have been to finals and we challenge.
"Everybody knows Everton don't have a big squad but that hasn't stopped him doing a great job here. Good luck to him."