The Daily Mail says the 21-year-old was David Moyes's first official signing from Penarol for £1.75m but Madrid want him to join up with Zinedine Zidane and their B team Castilla.
Real have asked for a loan for the right-back which could then become a permanent £4m deal.
Varela is capped at youth level for Uruguay, having played for both their Under 17 side and also the Under 20 side.
The former Penarol man has failed to make a single appearance for Manchester United's first team but has been used regularly for the club's Under 21 team.
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