The Armenian has fallen out of favour under Jose Mourinho, having struggled to apply his game under the manager's approach. Reports of a disagreement between the two have more recently emerged.
It is understood United would now be willing to let him go for the right deal, and Dortmund have so far shown the strongest interest.
That stance would be in line with the Bundesliga club's recruitment history as they have regularly shown a willingness to bring back former players who have struggled elsewhere, such as Mario Goetze, Nuri Sahin and former United player Shinji Kagawa.
Such a deal might also suit Old Trafford's plans for incoming players, as Mourinho does want a defensive midfielder, and Dortmund's Julian Weigl has long been a target. While any kind of direct swap is unlikely, a move could facilitate negotiations.
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