The Uruguayan right-back joined United from Penarol in the summer of 2013. He then spent time on loan at Real Madrid's Castilla under Zinedine Zidane's tutelage before returning to Old Trafford and making his debut under former Red Devils manager Lious van Gaal.
Eintracht boss Niko Kovac led Die Adler to a 16th-place finish in the league last season and will be hoping that the tireless fullback will help shore up a defence that conceded 52 goals.
Varela scored the last-minute title winner for United's Under-21 side last season against Tottenham Hotspur and went on to appear 10 times for the first team.
After his impressive breakthrough season, the 23-year-old received his first national team call-up for World Cup qualifiers against Brazil and Peru in March this year.