Rooney was among three players who suffered injuries in training on Friday, while the Reds' plans were further dented when Javier Hernandez sustained a dead leg after just three minutes at the Britannia Stadium.
"Jonny Evans got an ankle injury in training yesterday," Sir Alex told MUTV. "We had a nightmare of a training session - Evans, Michael Carrick and Wayne Rooney all got injured. Jonny thought he'd be okay but in the warm-up it was no use. With Chicharito in your team he gives you that penetration and speed, that was a loss."
"It's difficult to say [how long Rooney will be out for]. It's not a serious hamstring injury but I don't think he'll be available for next week. I think he'll be out for more than a week."
Rooney, Carrick, Evans and Hernandez join a lengthy injury list which already includes Nemanja Vidic, Chris Smalling, Rafael, Tom Cleverley and Darron Gibson, although Rio Ferdinand and Danny Welbeck both returned to Premier League action against Stoke.
"We are Manchester United and we don’t like to make excuses but it is difficult to come to a place like Stoke without those types of players," admitted Patrice Evra. "We have many players who are not here tonight."