Sanchez stormed down the tunnel after Chile's 1-0 defeat to Argentina on Thursday night.
Barcelona star Lionel Messi grabbed the only goal of the game, converting a first-half penalty.
Sanchez, who has been said to have thrown tantrums at Arsenal this season, was furious after the game.
Alexis Sanchez was fuming after Chile's 1-0 defeat to Argentina
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Lionel Messi struck as Argentina beat Chile on Thursday
But Chilean outlet Emol claim his compatriots are equally angered by his response at the final whistle.
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Players had urged the forward to walk over to the fans and thank them for their support.
But the Arsenal star ignored them and exited the pitch immediately after the final whistle.
Alexis Sanchez has cut a frustrated figure for club and country this season
A source reportedly said: "[We] made him see that they understood his anger, but that people deserved at least a farewell as a thank you for their support.
"There were no confrontations or anything, but a demonstration that fans are not to blame for the defeat."
Sanchez threw a strop after being substituted against Swansea in January and also stormed out of Arsenal training earlier this month.