The 20-year-old Uruguayan is moving to Old Trafford from Atletico Penarol, with United's official website reporting an undisclosed fee.
Moyes, who is new to the job after replacing the retired Sir Alex Ferguson, will be able to view Varela in action immediately as he is in Uruguay's squad for the Under-20 World Championships in Turkey.
Varela, who played just one senior game for Penarol, is a full-back and will provide competition for the likes of Rafael and Fabio da Silva, Patrice Evra and Alexander Buttner.
Moyes does not officially take up office at Old Trafford until July 1 but has been familiarising himself with the club ever since his departure from Everton was confirmed last month.
- Sports & Recreation
- Sir Alex Ferguson
- Old Trafford