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Varela, who had a two-week trial with United in May, will play for Uruguay at the Under-21 World Cup in Turkey which begins on June 21.
“I’m very pleased to be a part of this club, one of the best teams in the world,” Varela told the club's official website.
“As everyone in the world knows, this is a huge club that has won everything and I really hope that continues.
“I’m very lucky there are quite a few Spanish-speakers here. It will certainly help me settle in.”
The right-back shone for Uruguay's Under-20 side last season, and you can read more about Varela in Goal's profile of United's latest signing.