The 28-year-old is believed to have been dating actress Mayte Rodriguez for several months, and unhappy Chileans believe that she is responsible for making him "fat and lazy".
Sanchez has endured a tough time in recent weeks, having missed out on a £70m move to Manchester City and being part of the Chile side that suffered defeats to Paraguay and Bolivia during the international break.
Those losses have left the county outside of the qualification places for the World Cup with just two matchdays remaining and a protest has now been arranged for October 5, the day that Chile host Ecuador in Santiago.
The Facebook event is titled "March so that Alexis ends it with Mayte and returns to his level to reach the 2018 World Cup in Russia" and already has more than 9,000 interested parties.
Earlier this week Sanchez took to social media to hit out at "bad people [who] criticise without knowing".