Tutto Mercato Web claims the French tactician is an admirer of the 22-year-old who scored a brace in Japan's 3-0 victory over South Korea on Wednesday.
It is understood Wenger is interested in the Dortmund midfielder following speculation which claims that Cesc Fabregas and Samir Nasri could soon leave the London club.
Kagawa has previously indicated he is content with the German champions who will be competing in the 2011-12 Uefa Champions League.
However, the former Cerezo Osaka man, who has been linked with English champions Manchester United, had admitted in June he wants to play in the Premier League at some stage.
"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."
Arsenal already have one Japanese player on their books in teenage winger Ryo Miyaichi and previously had Junichi Inamoto at the club.
Kagawa, who missed a large chunk of last season due to a foot injury, made 18 Bundesliga appearances for Dortmund in the 2010-11 campaign, scoring eight goals and providing eight
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