Both players picked up injuries in United's 1-0 win over Norwich on Saturday, but Louis Van Gaal remains optimistic.
Martial was withdrawn before the 1-0 win over Norwich City after suffering an injury to his right leg during the warm-up, while Darmian later limped off with an ankle problem.
The news took the gloss of a victory that brings the Red Devils back into contention for Champions League qualification.
But Van Gaal remains optimistic about the duo's fitness. Explaining his decision to withdraw Martial, he told the club's official website: "When a player says he feels something, he shall not play. I think you have to be fit - especially in this weather as that is a big difference. We have to see tomorrow (Sunday) if the muscle is tight."
The Dutchman remains similarly optimistic over Darmian, adding: "Darmian has a kick on the ankle so we will have to see the reaction tomorrow (Sunday).
"It is too early to say [how serious it is]. Now he is walking in a brace but, tomorrow, it can be good again because a kick is a kick."
Nevertheless the news will interest Crystal Palace boss Alan Pardew, whose side take on United in the FA Cup Final at the end of May in a repeat of the 1990 showpiece.
The Eagles continued their preparations for that game at Wembley with a 2-1 win over Stoke City on Saturday.