In a statement, Elliott said he is “truly sorry” and the contents of the video “do not represent who I am as a person or how I’ve been brought up”.
“I realise that my actions were both immature and senseless,” he said.
“The video was taken whilst messing around with friends in a private environment and was not directed at any individual.”
He added: “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 midfielder, who made his professional debut aged 15 years and 174 days, joined Liverpool from Fulham earlier this week.