Following La Albiceleste's Copa America triumph earlier this month, the 23-year-old shared a video of himself and some of his teammates singing a derogatory song about French players of African descent.
Fernandez is reported to have apologised in person to his teammates in Atlanta on Monday, where the Chelsea squad is preparing for a friendly against Mexican side Club America on Wednesday night.
It is understood the Argentine has voluntarily offered to make a contribution to an anti-discrimination charity in recognition of his misjudgement.
The club has also privately pledged to match his donation through their charitable organisation, the Chelsea Foundation.
Fernandez not out of the woods just yet
BBC Sport report that several players appreciated Fernandez's attempts to apologise, and new manager Enzo Maresca is expected to announce the imminent return of the £107m midfielder.
In a media conference last week, the Italian downplayed the situation, stating that he did not anticipate any problems upon the 23-year-old's return.
"I don't think so," the Italian responded when quizzed about the issue on July 24. "They are all human beings. I don't think there are any bad intentions from any of them.
"I don't think when Enzo is back that we will have any situations. Enzo clarified the situation, the club the same, so there is nothing to add.
"They are not bad people or human beings. It can happen but I don't think there are any problems at all. I spoke with Enzo and all of them."
FIFA is currently investigating the matter, which might lead to involvement from the Football Association (FA) and could result in a potential ban from domestic matches for Fernandez.
Mixed reactions toward Fernandez among Chelsea players
Chelsea's first-team squad includes several French players who are Black or of mixed race: Axel Disasi, Benoit Badiashile, Lesley Ugochukwu, Christopher Nkunku, Malo Gusto, and Wesley Fofana.
Fofana, who has one cap for France, posted an image of the video on Instagram, calling it "uninhibited racism."
Blues' winger Noni Madueke was one player who supported his manager's view, suggesting that the situation would ultimately "be fine."
However, club captain Reece James described the situation as a "problem" and offered to step in as a mediator, along with five-cap France international Disasi.
The social media video also triggered a diplomatic incident, resulting in the dismissal of ministers in Argentina and prompting the French Football Federation to file a legal complaint with FIFA.