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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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.