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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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.