"Harvey Elliott has been suspended from playing in all domestic club football for a period of 14 days, running up to and including 24 October 2019, after admitting a breach of FA Rule E3 in relation to a video posted on social media and providing a public apology", the FA said in a statement.
In the video recorded during the Champions League final between Liverpool and Tottenham in May, Elliot is heard calling Kane a "fucking mong", which is a slang word that is short for the politically incorrect term "mongoloid".
The 16-year-old, who admitted breaching FA rules, must also complete an education course and pay a £350 fine.
He later issued an apology on social media, saying he was "truly sorry".
"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".
A Liverpool spokesperson said: "Harvey's apology was honest, immediate and unequivocal".
Liverpool confirmed the signing of Elliott on July 28 after he turned down the opportunity to remain at Craven Cottage, with his youth contract expiring a month earlier.
Elliott has played one match for Liverpool's senior team since his move, in the English League Cup last month.
He was named in the squad as Jurgen Klopp's side extended their flawless start to the league season with a 2-1 win over Leicester City.