The 28-year-old was the victim of a strong tackle by West Brom's James McClean during the Gunners' 3-1 defeat at the Hawthorns on Saturday and failed to finish the game, being replaced late on.
After the match, boss Arsene Wenger said Sanchez's ankle was in "an absolutely terrible state" and feared the club's top-scorer had suffered ligament damage.
Arsenal wanted to carry out further tests on Sanchez's injury before deciding whether he could join up with the Chilean national team for their World Cup qualifiers against Argentina and Chile.
However, according to reports from South America, Sanchez has already left for Chile and will undergo an assessment on his ankle when he arrives on Sunday.
Wenger was unhappy with how the Chilean medical team treated his star forward during November's international break, allowing Sanchez to play despite him carrying a hamstring injury, and the Frenchman will be worried the former Barcelona man could play through the pain barrier once more.