Elliott became the youngest player to feature in the Premier League when he came on as a substitute against Wolverhampton Wanderers in May, aged just 16 years and 30 days.
He cannot turn professional until his 17th birthday next April, but it is understood he will reject a scholarship at Craven Cottage and move to the Champions League winners.
If, as expected, Liverpool win the race to sign Elliott, they will have to negotiate a compensation package with Fulham as the England Under-17 international is regarded as a highly prized prospect.
Elliott, who also attracted attention from Arsenal and Manchester City, will become the second emerging talent to join Liverpool this summer after 17-year-old Dutch defender Sepp van den Berg moved from PEC Zwolle.
Liverpool paid an initial £1.3m for Van den Berg, which could rise to a maximum of £4.4m depending on add-ons.