Kagawa has played just seven minutes in the Community Shield for United so far this season, having not made a single Premier League appearance, but said he hoped for chances after scoring for Japan during the international break.
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"There are quite a lot of players who have not featured yet," Moyes told reporters. "But if you look at it most of them were late back from international duty after the Confederations Cup.
"Then they were away for a friendly international and a long-haul flight, then they were away again for another international.
"If you look at the reasons why people haven't played, it is more to do with them not being available for that to happen."
Moyes confirmed on Friday that he was pursing a double swoop for Leighton Baines and Marouane Fellaini until late in the transfer window - eventually signing just the midfielder in a £27.5 million deal.
And the new manager has insisted his moves for a new left-back - including a late attempt to sign Fabio Coentrao on loan from Real Madrid - was purely to provide cover for Patrice Evra, although he admits he is yet to talk to his captain regarding his plans for the season.
He added: "I just felt we needed to make sure we had good cover for Patrice. He has played brilliantly well but I hope we are going to have a season that is going to include 60 or 70 games.
"Patrice played a big majority of the games last season but I just want to make sure if he broke down I had the right people around to help me.
"I have not because he is such a good professional. He knew exactly what we were doing. If you play for Manchester United there is always someone out there getting ready to take your jersey. It is up to you to fight and make sure you keep it."