The London Evening Standard says the World Cup-winning midfielder, 27, has one year left on his contract in Spain and is refusing to sign a new deal at the Bernabeu.
The European champions will not match his £200,000-a-week wage demands, and Carlo Ancelotti confirmed yesterday that Khedira had rejected their latest contract offer.
Both Arsenal and Chelsea - whose manager, Jose Mourinho, has a good relationship with Khedira from their time in Madrid - have shown an interest but it is understood that United have moved to the front of the queue.
Unlike the Gunners, United manager Louis van Gaal would be prepared to meet Khedira's salary requirements and the fee of about £12million - relatively low because the Germany midfielder could go for free next summer - would be no trouble to the Old Trafford club.
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