Enzo posted a video to social media of Argentina's players singing a song aimed at their World Cup rivals France, targeting the make-up of the Les Bleus squad.
The Argentines sung how French players are "from Angola" or have a "Nigerian" mother and a "Cameroonian" father.
In reaction, Chelsea defender Wesley Fofana posted to X: "Football in 2024: uninhibited racism".
It then emerged that Fofana, Axel Disasi and Malo Gusto had all unfollowed Enzo on Instagram.
Chelsea have now released a statement, confirming an "internal investigation" is underway.
The club announced: "Chelsea Football Club finds all forms of discriminatory behaviour completely unacceptable. We are proud to be a diverse, inclusive club where people from all cultures, communities and identities feel welcome.
"We acknowledge and appreciate our player’s public apology and will use this as an opportunity to educate.
"The Club has instigated an internal disciplinary procedure."
As stated, later on Tuesday evening, Enzo posted an apology, stating: "I want to apologise for a video posted on my Instagram channel during the national team celebrations.
"The song includes highly offensive language and there is absolutely no excuse for these words. I stand against discrimination in all forms and apologise for getting caught up in the euphoria of our Copa America celebrations.
"That video, that moment, those words, do not reflect my beliefs or my character. I am truly sorry."