Moyes' predecessor Sir Alex Ferguson was renowned for his toughness, handing out frequent "hairdryer treatments" to those who were not meeting his demands.
Fellaini - who played under Moyes for four years at Goodison Park - says that any United player hoping for a more relaxed atmosphere around Old Trafford will be disappointed.
The Belgium international - who is being linked with a summer move to United - discovered first hand what happens when Moyes is upset when he was suspended for headbutting Stoke's Ryan Shawcross.
"He is a decent man, a great manager, but believe me he can be tough. If you want to win something, to create a good team you have to be hard sometimes with the players - football is like this," Fellaini told reporters. "He can be scary.
"I remember when I was banned after the Stoke game, he gave everyone the day off - everyone: The players, the entire staff - but I came to the training camp to do a training session with him.
"It was just me and him, and it is 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 hard!
"He put me through a tough session... it was a full-on session, running, shooting, passing, everything and very intense, and afterwards I was upset.
"I said afterwards, 'Arggh, this is bad', but I am happy when he looked after me, so I understand."