Elliott, who is still just 17, has spent this season on loan alongside Downing at Blackburn Rovers - notching 11 assists while scoring five for the Championship club and playing a starring role.
Former Reds winger Downing told Goal: "From what I've seen up to now - he's going to be in the Premier League for sure.
"I know [Bournemouth manager] Jonathan Woodgate well, and he's quite close to [Liverpool sporting director] Michael Edwards. He asked me about Harvey when he came to Blackburn, and I was like 'f***ing hell, he's brilliant!'
"It's his game-understanding, when to pass the ball, when to dribble, when to come inside, when to stay wide. For someone so young, it's top-drawer.
"Woody told me they have high hopes for him at Liverpool, and I can see why.
"But he knows better than anyone that he has a big job on his hands to get into the team.
"It's one thing doing it at Blackburn, but we are talking about a team that wants to win Premier Leagues and Champions Leagues. Liverpool have to challenge, and Harvey has to get Mo Salah out of the team!
"But things can happen. If Salah leaves or gets injured, then he may get a chance, and then it's up to him to take it.
"He's got the quality, and playing with better players will bring the best out of him. And Klopp obviously likes him, so he has a chance."