Sanchez, 30, injured the medial ligament in his right knee during United's 3-2 Premier League victory over Southampton on Saturday. However, he is expected to be fit to represent Chile at the Copa America, which starts on June 14, national team physio Pedro Onate told Chilean newspaper El Mercurio.
Sanchez could return for United's last two league games against Huddersfield Town and Cardiff City, but any setback in his rehabilitation would mean he has played his last match this season.
United are already without a host of players for Wednesday's clash at PSG due to injury, while midfielder Paul Pogba is suspended for the tie.
They trail 2-0 on aggregate after losing the first leg at Old Trafford.
(Reporting by Simon Jennings in Bengaluru; Editing by Ian Ransom and Christian Radnedge)
