He pulled up chasing a long ball in the third-place play-off against Argentina, hobbling off after 17 minutes.
The 30-year-old, who has endured a difficult time at Old Trafford since joining from Arsenal in January 2018, is not expected to feature on the club’s pre-season tour.
United set off for Australia on Sunday.
Sanchez’s injury came in a thrilling game, which Argentina won 2-1 despite having captain Lionel Messi sent off.
United begin their Premier League campaign against Chelsea on 11 August.