The Guardian says Elliott, 16, has joined the Reds from Fulham under freedom of contract.
However, the Anfield heavyweights are yet to make a compensation offer to the west London outfit.
Reds boss Jurgen Klopp wasn't the only Premier League boss battling it out for his signature.
It's suggested Ralph Hasenhuttl was also vying for Elliott's services.
Last season, the exciting attacker became the youngest player ever to feature in a Premier League fixture.