The video, which was shared privately on Snapchat and includes an offensive caption, appears to show Elliott, a Fulham player at the time, imitating Kane and using derogatory language towards the striker.
The video appears to be filmed by Elliott following the Champions League final on June 1, in which Liverpool defeated Kane's Spurs 2-0.
In a statement released on Wednesday, Elliott, 16, said: "I would just like to wholeheartedly apologise for any offence caused on the back of a video of me circulating on the internet.
"The video was taken whilst messing around with friends in a private environment and was not directed at any individual but I realise that my actions were both immature and senseless.
"I would like to stress that the contents of the video do not represent who I am as a person or how I've been brought up, and I truly am sorry."
Elliott, who has supported Liverpool since he was a boy and completed a move to the European champions last week from Fulham, had made his second appearance for the club in their preseason win over Lyon earlier on Thursday.
He is yet to sign a professional contract with Liverpool and is unable to do so until his 17th birthday, which is in April next year.
Liverpool are aware of the incident and have made clear to Elliott that his conduct in the video is unacceptable. They have also warned Elliott that a repeat of such behavior while he is at the club will result in disciplinary action.
Liverpool also released a statement on Thursday.
"The club believes Harvey is right in offering a full and unreserved apology for the comments made in the video," a club spokesperson said.
"Clearly the use of such language is completely wrong and not in keeping with our values or what we would expect from any employee.
"Given the clip was recorded prior to the player signing for the club and was not in the public domain at that time, we won't be making any further comment publicly."
Elliott is the youngest player to ever play in the Premier League having broken the record in May 2019 aged 16 years and 30 days. He made his debut for Fulham in the Carabao Cup aged 15.
Sources have told ESPN that Paris Saint-Germain, Real Madrid, Barcelona and Borussia Dortmund were all interested in signing the England youth international, prior to his move to Liverpool.