The Mexico international has been in fine form for David Moyes' side with seven goals in La Liga, but will now face some time on the sidelines.
Vela was substituted after 17 minutes of Sociedad's 4-1 defeat to Real Madrid at Santiago Bernabeu and scans have shown damage to his knee.
"During the match held in Madrid he twisted his right knee," a club statement read. "The examinations identified a torn medial meniscus.
"After performing specialised consultations to assess the most appropriate treatment, we opted for surgical treatment.
"Carlos will be operated today [Monday], after the surgery a new medical report will be issued."
Without Vela in attack, Moyes will be forced to rely on Imanol Agirretxe and Alfred Finnbogason to keep Sociedad moving forward.
However, the pair only have two goals between them in 30 league appearances this season.
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