Lingard pulled up shortly before half-time at Old Trafford with what appeared to be a hamstring injury.
He played on after receiving treatment but was replaced by Alexis Sanchez during first-half stoppage time.
Martial was injured earlier in the game and played on until the interval but did not reappear for the second half. His place was taken by Juan Mata.
The two injuries will be a major worry to Ole Gunnar Solskjaer with a crucial run of games in the next few weeks.
United travel to Chelsea in the FA Cup fifth round next Monday before hosting Liverpool on February 24.
Games against Crystal Palace and Southampton follow before the away leg against Paris Saint-Germain on March 6.