The home grown talent was seen ignoring a ball boy in a clip that has gone viral on social media.
However, the Blues believe the footage is taken out of context and does not show the entire incident.
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In a statement released on Wednesday afternoon, the club said: "Chelsea Football Club is aware of a video clip circulating on social media from Saturday's fixture against Burnley, which has been taken considerably out of context.
"The subsequent level of abuse and defamatory comments directed towards Conor Gallagher is completely unacceptable.
"We are proud to be a diverse, inclusive club where people from all cultures, communities and identities feel welcome."