The draw for the tournament was made on Wednesday morning.
United will be at home to Real in a single-legged knockout tie which will be held at the start of March. It could mean a reunion for United wing-back Alvaro Fernandez, after his move from Real Madrid last summer.
Ties for the round of 64 are set to take place on either Tuesday 2 March or Wednesday 3 March.
United qualified for the Youth League as a result of the senior side's qualification for the Champions League tournament last season.