Former Paris Saint-Germain player Menez suffered the injury during his side's friendly against Pachuca last week and will undergo surgery in France "as soon as possible."
Menez signed for Club America on a one-year deal from Turkish side Antalyaspor in early January and helped them to a second-placed finish in the regular season in the 2018 Clausura.
Meanwhile, former Club America defender Diego Reyes was at the club's training facility in Mexico City on Saturday.
The Mexico international, who missed the World Cup due to injury, is continuing his search for a new club after his contract with Porto expired.
Club America face Morelia in San Jose on Sunday ahead of the Liga MX Apertura opener against Necaxa on July 22.