Elliott was not a professional player when he moved to Anfield due to his age, with Fulham demanding around £8million for the player. Liverpool claim they are prepared to spend £750,000, says the Daily Mail.
Teams who sign footballers aged under 24 must pay a compensation fee if they secure the player while he is not on professional terms or once his contract comes to an end.
Elliott has made nine senior appearances for Jurgen Klopp's side since his arrival in Merseyside, with eight of them coming last season.
The 17-year-old made six cup appearances last term and appeared twice in the Reds' Premier League title-winning season.
The winger was sent on loan to Championship side Blackburn for this season.