The former X Factor popstar, who is a lifelong fan of the Premier League champions, presented the ex-Borussia Dortmund midfielder with a signed copy of his album.
"It was a pleasure to meet Shinji," Murs told the club's official website. "He knew my music and sang a bit of 'Troublemaker' which was great.
"He's a really good guy, and he gave me a few tips on what to say in Japanese."
Kagawa has struggled to secure a place in the United first team under manager David Moyes this season.