The 21-year-old was David Moyes's first signing as United boss, but has not featured in the first team since his move from Penarol and links up with the European champion for the remainder of the 2014-15 campaign.
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Varela will play for the Real's 'Castilla' reserve team in the Segunda Division under Zinedine Zidane.
He becomes the second player this week to make a loan move to Real from United, following striker Javier Hernandez to the Spanish capital.
Uruguay Under-20 international Varela was a regular fixture in United's Under-21 team last season.