The Brazilian left-back sustained the problem during the club's training camp in Surrey this week and has returned home, missing Saturday's friendly with QPR.
Boss Ole Gunnar Solskjaer, who signed a new contract, said on the club's official website: "Unfortunately, Alex slipped and he twisted his ankle so he'll be out for a few weeks.
"We're hoping that it's not going to be too bad but he'll be out for a little while."
Telles looks set to miss next week's friendlies against Brentford and Preston, with another warm-up match against Everton on 7 August.
United kick-off their Premier League campaign against Leeds seven days later.