The 21-year-old Uruguayan was David Moyes' first signing as United boss, but has not featured in the first team since his move from Penarol last June and links up with the European champions for the remainder of the 2014-15 campaign.
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.
Alexia Putellas: "We’re already the only club to have won the men’s and women’s Champions League, so the only thing even more historic to do than that would be to win it in the same season"
‘Trent has won everything he can for Liverpool, but of those who’ve left that we wanted to stay, who’s actually done better? I can only come up with one’: Reds legend highlights potential danger of Alexander-Arnold moving to Real Madrid
