The 22-year-old has been one of the best performing Asian players in Europe last season, helping Dortmund to their first Bundesliga title in nearly a decade.
And Kagawa once again impressed on Wednesday after scoring a brace in Japan's 3-0 win over Korea Republic in a friendly.
According to Tutto Mercato Web, Wenger is a huge admirer of the former Cerezo Osaka man, who has not ruled out a move to the Barclays Premier League.
"If I get a formal offer, I'll be strongly interested in listening to what they have to say," Kagawa told Nikkan Sports.
"The Premiership is one of the leagues I want to challenge. If it is Man United, I'll be still more eager to go and try how much I can do there."
Wenger is known to possess a liking for Japanese players, having previously signed Junichi Inamoto and Ryo Miyaichi, who has recently received his work permit to play for the Gunners.