The former Real Madrid, Roma and Valencia player suffered a "detachment in the graft area" of his right knee, the club said, and was forced to stop training.
Gago, 31, has injured the knee three times in three years but was striving to be fit in time for the tournament in Russia.
This new setback will keep him out for between two to three months, a club source told Reuters.
Argentina kick off their World Cup campaign on June 16 against Iceland and then face Croatia and Nigeria.
(Reporting by Ramiro Scandolo, Writing by Andrew Downie; Editing by Christian Radnedge)