FFF president Philippe Diallo "condemned with the greatest firmness the unacceptable racist and discriminatory remarks made against players of the France team".
The chants were heard during a live video posted on social media by Chelsea and Argentina midfielder Enzo Fernandez from the team bus in the wake of the Copa victory over Colombia in Miami on Sunday.
Some players, including Fernandez, sing a chant dating back to the 2022 World Cup final that Argentina won against France.
The song targets France's star striker Kylian Mbappe and includes homophobic insults.
"Faced with the seriousness of these shocking remarks, contrary to the values of sport and human rights, the president of the FFF decided to directly challenge his Argentine counterpart and FIFA and to file a legal complaint for racial and discriminatory offensive remarks," an FFF statement read.
French Sports Minister Amelie Oudea-Castera also called for world football's governing body FIFA to take action.
"Pathetic. Behaviour all the more unacceptable because it is repeated," Oudea-Castera wrote on X.
"FIFA: a reaction?" she added.
Argentina and France have maintained a heated rivalry since the South Americans last 16 elimination to eventual champions France at the 2018 World Cup.
Two years ago, the FFF also filed a complaint against racist comments on social media.
