The Daily Star says United have been offered Di Maria for £40m.
Di Maria helped Real win the Champions League last season and scored Argentina's extra time winner against Switzerland at the World Cup on Tuesday.
But talks about a new, improved contract for the 26-year-old have stalled amid claims in Madrid he is being too greedy, and Real have decided to test the market for him.
United have been keeping tabs on Di Maria but value him at closer to £30m.
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