Aubameyang has got 21 goals in 21 Bundesliga games this season as they bid to keep up with Bayern Munich.
The 26-year-old joined Dortmund from St Etienne in 2013 and was crowned African Player of the Year for 2015.
A move to either club this summer would smash their transfer records with United's standing at £59.7m for Angel Di Maria and Arsenal's at £42.5 for Mesut Ozil.
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