The 16-year-old has emerged as a target for the Champions League winners over the past couple of weeks.
Reports in England last week suggested the Reds were leading the race to sign the Fulham midfielder.
Elliott became the youngest-ever player to feature in the Premier League last season.
However, the England Under-17 international was powerless to help Fulham remain in the Premier League.
The former Queens Park Rangers youth player finished his first season in the first-team with three appearances in all competitions.
Elliott is thought to be attracting interest from La Liga giants Real Madrid as well as Ligue 1 champions Paris Saint-Germain.
However, former Fulham and Liverpool FC midfielder Murphy would like to see Elliott complete a switch to Anfield.
“Harvey Elliott would be a great signing for Liverpool and [it is] the right move for the player,” Murphy told bWIN.
“Liverpool are in a very stable position with Klopp and as a youngster, Elliott wouldn’t have to worry about another manager coming in and not seeing him as part of his future plans.
“Klopp has a great record of giving younger players a chance to prove themselves and I’m sure Elliott will get the same opportunity if he buys in to Klopp’s plan.
“This would give him a great chance to play and train with some of the best players in the world and see what it takes at the top.
“He’s very young and has the chance to develop at Liverpool and then spend a season out on loan if need be. It’s a great opportunity for the player and one I can’t see him passing up on.”
Elliott made his Premier League debut in a 1-0 loss to Wolves, becoming the youngest-ever player to feature in the English top flight at the age of 16 years and 30 days.
The Reds have already signed 17-year-old Dutch talent Sepp van den Berg from FC Twolle this summer.
Liverpool FC won the Champions League for the sixth time last season thanks to a 2-0 victory over bitter rivals Tottenham Hotspur in Madrid at the start of June.
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