The Guardian reports that Moyes is a long-time admirer of the Croatian midfielder.
Real Madrid are reportedly prepared to sell the 27-year-old if United make a bid in the region of £25million for the midfielder.
Modric joined Real Madrid from Tottenham for £33million last summer, but failed to impress under Jose Mourinho at the Bernabeu.
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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
Who of Europe’s top strikers will move to England this summer?