Kagawa linked up with his United team-mates for the first time since May on Monday, when the squad arrived in Japan on the third stop of their pre-season tour.
And although the 24-year-old is not expected to feature in Tuesday's game against Yokohama F Marinos, Moyes says he is looking forward to learning more about Kagawa this season.
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"Sir Alex spoke in glowing terms about Shinji and how good a player he is. I've just met him today, so we're getting to know each other. I'm trying to learn Japanese and he's trying to understand my Scottish.
"It was his first season in the Premier League last year. He's a young player so we hope his development continues. I look forward to working with him and hopefully he has a really good season coming up."
The Japan international featured 26 times for the Premier League champions in his debut season, scoring a total of six goals, but recently admitted to feeling lost at Old Trafford.
Nevertheless, Kagawa is hoping to make a quick impression and build a strong working relationship with his new manager.
"We still need more time to get to know each other, but I'm looking forward to building up our relations," he said.
"As a player, I feel every game is important and I want to give my very best to give a good impression to the coach."