The Germany international was linked with the Gunners over the summer, before Arsene Wenger eventually completed a £42.4m deadline day deal for Mesut Özil.
That effectively ended Arsenal's interest in the 22-year-old Gundogan, but the report claims United boss David Moyes is now weighing up a move for the highly-rated midfielder in the January window.
Moyes acted late over the summer to bring in Marouane Fellaini, but only after moves for Cesc Fabregas, Sami Khedira and Ander Herrera had failed.
And, despite the addition of Fellaini, it seems the United manager is still on the lookout for a world class midfielder to add to his ranks.
The Old Trafford club are short of options in the middle following injury to the Belgian international, with Michael Carrick, Anderson and Tom Cleverley the only first-team players available. In contrast to United, Arsenal have several options in that particular area of the pitch.
Gundogan is under contract until the summer of 2015, and Dortmund are believed to have offered him an extension in a bid to ward off suitors.
Regarding contract negotiations, Gundogan said last month: "I am not going to make statements on the contents of those talks.
"Everyone has to be patient about it, just like I have to be patient in terms of my current injury."
The young midfielder has featured in just one game for Jurgen Klopp's side this term having picked up a back injury playing for his country.
He's rated at £30m, but if he stays at the club in January and fails to sign a new deal, he'll have just one year left on his contract next summer and Dortmund may struggle to command such a large fee.