The Belgium international suffered an ankle injury in a controversial challenge with Liverpool's Sotirios Kyrgiakos in the Merseyside derby defeat at Anfield on Saturday.
It is feared the 22-year-old could be out for a few weeks - and he has definitely been ruled out of Wednesday's Premier League clash with Chelsea - but Everton are still to learn the full extent of the problem.
Moyes said: "He was due to see a specialist in London but the train he was due to travel down on was cancelled, so he's missed his appointment time. He's booked in now for Thursday. But he's got a really bad ankle. I was really surprised - it was like an elephant's ankle, it was that bad.
"The boy has got a bad-looking injury but until the specialist sees it we can't make any kind of decision. He has been integral part of how we have played in recent weeks and he will be a miss.''
Fellaini's injury at least comes at a time when Everton have the strength to cover. The Toffees were hampered by injuries throughout the opening half of the campaign but Moyes spoke positively last week about how close he was to having a fully-fit squad.
Mikel Arteta, only recently returned from close to a year out with a knee problem, replaced Fellaini at the weekend and could continue after coming through unscathed. Fellaini was hurt as both he and Kyrgiakos attacked a loose ball.
Kyrgiakos was sent off for his part in the tackle but some observers have claimed Fellaini was equally to blame for the collision and argued that either player could have seen red. But Moyes said: "I said it after the game, and I haven't changed my views, that it was a two-footed tackle off the ground (by Kyrgiakos). He was flying towards a player, he could see that coming all the way, so it was a sending off.''
The Fellaini-Kyrgiakos clash was not the only flashpoint in a bruising derby encounter, which Liverpool edged 1-0 with a Dirk Kuyt goal. Everton's Steven Pienaar was also sent off for two bookable offences and Moyes admitted afterwards the South African could have been dismissed earlier than he was.
There were also a number of other incidents but Moyes insisted there was no deliberate plan from either side to play so physically. He said: "Nobody sets out for it to be like that but they sometimes turns out that way.''
The derby defeat ended Everton's recent upturn in form but Moyes expects his side to recover quickly. He added: "We don't really need to dust ourselves down. We went nine unbeaten before it and we've lost a game to a set-piece.
"We had an awful lot of the ball and there are a lot of positives to take. We are not going to get too down about it. We are going to try to kick on and win the next game against Chelsea.''
Everton drew 3-3 with the leaders when the sides met at Stamford Bridge in December and Moyes sees no reason why his men cannot upset the title challengers again. Moyes said: "I think we are playing well and don't have anything to fear, but we respect Chelsea because they have really good players.'' Pienaar misses the game at Goodison Park through suspension.