According to the Daily Mail, Arsenal officials have 'made contact' with Gundogan's family ahead of a potential switch to the Emirates Stadium this summer. The player's father acts as his agent.
The reason why they've moved so early is because Manchester City and Manchester United are both thought to be planning bids for the gifted 24-year-old as well.
Bayern Munich and Atletico Madrid are also interested in £20m-rated Gundogan, although Dortmund claim they are yet to receive any offers.
The Germany international's chequered injury history may put off one or two suitors, but clearly Arsenal are not one of them. Arsene Wenger wants to strengthen his central midfield, with the Gunners weak in this position.
Mikel Arteta is no longer the answer in central midfield, and injury to Jack Wilshere has restricted his effectiveness even when fit.
Francis Coquelin has emerged as a useful option this season, but a player of Gundogan's class is on a different level to the Frenchman, something that will be crucial if Arsenal plan to break into the latter stages of the Champions League.
Regarding interest from Manchester United and Manchester City, both clubs have other first-choice targets but are monitoring interest in Gundogan as a back-up option.
City are still pursuing Paul Pogba of Juventus while Louis van Gaal remains a fan of Paris Saint Germain's Italian international Marco Verratti.
Both players are potentially more difficult targets though, with Gundogan widely expected to leave Borussia Dortmund at the end of the season, following the club's coach Jurgen Klopp.
What do you think Arsenal fans - would Gundogan be a good signing for the Gunners or are you worried about his injury record?
Have your say below!