The 20-year-old has signed a five-year contract, but is willing to move away temporarily in a bid to gain experience with Boca Juniors a touted destination
Varela, who has signed a five-year deal, is David Moyes' sole new arrival this season, with Wilfried Zaha's move completed in January, after 20-year-old defender impressed United's scouts while playing for Penarol.
Boca Juniors have been touted as a possible destination for the Uruguayan, and the defender admitted that he would welcome the move to La Bombonera.
"It could be a good solution," he told Tutto Mercato Web.
"I would be able to play continuously for a year but I'd be always thinking about returning to United, who are a club that believed in me."
Varela was part of the Uruguay squad that reached the final of this year's Under-20 World Cup in Turkey, where they lost to France on penalties.