Lanzini suffered a knee sprain in Argentina's 0-0 draw with Colombia on Thursday in a warm-up match for the Rio Games and missed the 3-1 friendly win over Haiti in Miami on Sunday.
“The West Ham man will remain in Miami from where he will leave for London,” the AFA said in a statement.
The Argentina squad was set to leave for Puebla to play Mexico in a final tune-up game on Thursday.
Argentina had hoped Lanzini might recover quickly enough for the Olympic football tournament, which kicks off on Aug. 4, but want to start with 18 fit players so the 23-year-old will be replaced by Boca Juniors forward Cristian Pavon.
Lanzini would have missed the start of West Ham's Premier League season, which kicks off at Chelsea on Aug. 15, due to his involvement at the Rio Games but could now be available if he recovers from his injury in time.
His departure is a blow for twice gold medal winners Argentina, whose head coach Gerardo Martino quit three weeks ago, since they have struggled to put together a strong squad with some clubs refusing to release players.
(Writing by Rex Gowar; Editing by Ken Ferris)
