Moyes must now cope without the England international, who injured his shoulder when falling awkwardly.
"We've got a few injuries and one is (from) the challenge the keeper made on Ashley Young," Moyes told a news conference.
"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."
United's third round tie with Swansea at Old Trafford is followed 48 hours later by a League Cup semi-final, first leg away to Sunderland.
"It's the same for both clubs (United and Sunderland) so there can't be many complaints but I don't think it's right to be asking footballers to play on Sunday and then Tuesday," Moyes said, with Sunderland hosting Carlisle United in the FA Cup.
"They did it over Christmas and I totally understand that and the reasons why. But it's a bit harder to accept at such short notice here," he added.
Despite their hectic fixture list and the fact that United face a battle to climb into the top four in the league, Moyes said the FA Cup remained a priority, especially with the club having not won the competition since 2004.
"The Cup is important to me, it always has been," Moyes said. "If you're a club like United you want to be involved in all the trophy-winning opportunities, and to not win the FA Cup for 10 years is something that's a bit of a surprise."
(Reporting by Martyn Herman; Editing by Ken Ferris)
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