Fellaini missed Saturday's 2-1 win over Sunderland with ligament damage in his wrist, which is feared will keep the Belgian international sidelined for at least three weeks.
Moyes has revealed that Fellaini is booked to visit a hand specialist on Monday ahead of an assessment by his national team doctor.
He told The Mirror: "He's got a wrist injury. He's going to see a specialist. It doesn't sound too good but we'll wait and find out on Monday."
The 25-year-old is expected to miss Belgium's upcoming World Cup qualifiers, with a return unlikely ahead of United's next Premier League game at home to Southampton on October 19.