The America forward suffered a muscle spasm at the back of his right upper thigh on the 31st minute in Chile's 1-0 win over Ecuador.
Beausejour, nicknamed "El Frances" or The Frenchman, will be significant loss, as he has started in the team's all 17 matches this season, according to the team's website. He is still in Chileat present, although the statement said that America is in constant touch with the Chilean national team's doctors.
The Mexico City team says that final studies will be ready late on Thursday and will determine if Beausejour can return to Mexico or will have to stay in Chile until his treatment is completed.
America has already lost one of its top players for the Sunday match: Argentine Daniel Montenegro was ordered to miss a game after receiving a red card.
The team will face off against Puebla from the nearby state of the same name on Sunday.
America is eighth in Mexico's 18-team league in the Opening Tournament with 23 points.
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