The video originally posted on Snapchat went viral this week after 16-year-old Elliott, who became the youngest player to ever play in the Premier League last season, made his move to Anfield.
Elliott, who describes himself as a boyhood Liverpool fan, appears in the video using derogatory language apparently aimed at Kane during the Champions League final between Spurs and his new club.
Kane is shown at the beginning of the video on a TV screen during the game before the camera turns onto Elliott, who seems to be topless.
In a voice thought to be mocking Kane's, the teenager makes incoherent noises in an apparent attack on the striker's speech.
The youngster posted a statement on his Instagram account apologising for his behaviour.
He said: “I would just like to wholeheartedly apologise for any offence caused on the back of a video of me currently 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 of how I’ve been brought up, and I am truly sorry.”