The 28-year-old, who left Valencia to re-join his first club Boca Juniors in July, suffered a knock during a feisty 0-0 Primera Division draw against Colon.
During a match in which 11 players - including Gago, who played the full 90 minutes - were booked, the former Real Madrid man sustained damage that has seen him ruled out until the end of the domestic season.
A Boca statement said: "The midfielder suffered a grade two sprain in his left knee, with partial lesion of the collateral medial ligament, and will be immobilised for 15 days.
"The estimated recovery time is 60 days."
Two months on the sidelines would put Gago's involvement in this year's World Cup under question.
Argentina play friendlies against the Trinidad and Tobago and Slovenia in early June before their tournament gets under way against Bosnia-Herzegovina on June 15.
Gago, who usually operates in defensive midfield, is a key member of Alejandro Sabella's squad, and the news will come as a blow to Argentina's preparations for their bid for a third World Cup title.
As yet, Sabella has made no indication whether a replacement for Gago, who has won 47 caps for his country, will be called up.
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