The fixture was brought forward to this week after United advanced to the FA Cup quarter-finals.
The new date has left Zaha with little time to recover from a hamstring injury sustained early last month.
Asked if the exciting winger could play, Hodgson replied: "I doubt it, unfortunately. I think we have to wait.
"The Manchester United game, that has been thrust upon us, is obviously something we're pretty unhappy about, especially in this current state of affairs where we do have so many of our good players, our first-team players, unavailable."