The Armenia international is expected to be squeezed out of United next month after a bust-up with manager Jose Mourinho.
Inter Milan have been in contact, while BILD says BVB are also discussing the idea of bringing him back to Germany.
BVB see the addition of Mkhitaryan boosting their top four hopes as they battle for Champions League qualification. With the midfielder needing no time to adjust to the Bundesliga, BVB could push for a move in the coming weeks.
For his part, Mkhitaryan is keen to quit United, recognising he has little chance of a regular start under Mourinho.