The midfielder was hurt during a collision with Tottenham goalkeeper Hugo Lloris during Wednesday's 2-1 loss at Old Trafford, while Robin van Persie remains unavailable
The England international was hurt in a collision with Tottenham goalkeeper Hugo Lloris in United's 2-1 loss at Old Trafford, an incident which left Moyes bemoaning Howard Webb's failure to award a penalty.
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"We've got a few injuries and one is [from] the challenge the keeper made on Ashley Young, which is going to keep him out for a time," Moyes told reporters.
"It's a shoulder injury and we're not sure exactly how long he's going to be out yet. We've got some other injuries as well that we'll just need to look at and see how they are over the next few days.
"Robin's coming on but we'll let you know when he's ready."
Moyes once again reiterated his belief that Young should have had a penalty, but ruled out suggestions video technology should be used to help referee's spot contentious decisions.
He added: "I don't think we're going down the route of video technology, simulation should be one their [referees] allowed to view.
"I wouldn't want videos to come into the game. Goal-line technology is a good situation, it seems to be working well when it's had to work.
"We trust the referees to make the right decisions, like all humans they make wrong ones sometimes. I thought it was the wrong decision against Tottenham, it should have been a penalty."