The Daily Mail newspaper claims that a four-year deal has been struck with the 25-year-old.
However, the Gunners have turned their attentions towards the Switzerland international Granit Xhaka.
Gundogan has been ruled out of Euro 2016 and the beginning of next season after dislocating his knee during training earlier this month.
The German outlet Bild report that it will take Gundogan four to six months for him to make a full recovery.
The injury put his proposed move to the Premier League in doubt and caused Dortmund to postpone contract negotiations.
Dortmund reportedly gave Gundogan an ultimatum to either extend his current deal or leave the club for a transfer fee.
The Germany international of Turkish origin had a year remanning on his Dortmund contract and was reluctant to sign a new contract with the German giants.
Gundogan has seven assists and three goals in 40 appearances for Dortmund this season.