The 19-year-old midfielder has been a regular with Jurgen Klopp’s first team squad this season and scored his first Champions League goal against Ajax on Tuesday.
Anfield great Gillespie has been impressed with the teenager and feels he is currently the closest one to Trent Alexander-Arnold to come out of the Liverpool academy.
The Scot is eager to see if Jones can take the next step and establish himself as a mainstay in the Reds side, but is certain about his abilities, especially on the ball.
Gillespie is also of the view that Jones offers what others Liverpool midfielders do not with his keen eye for forward passes where others tend to look for the safer options.
“I think Curtis Jones is one of these youngsters that is probably ahead of the other ones“, Gillespie said on LFC TV after the win over Ajax.
“I think we have seen that, he has been a regular in the squad and I think we are just reaping the benefit of that.
“I think he is the next close one to Trent Alexander-Arnold.
“Whether he is able to take that next step and establish himself as a first team regular [remains to be seen].
“But I think his ability and calmness in possession of the football as well really stands him in good stead.
“When you look at him, he never gets fazed, he is always comfortable, he is always looking to let the ball run through him as well as a midfield player.
“I think that is a little bit different to a lot of the midfield players that we have, the likes of Gini Wijnaldum, maybe Jordan Henderson and certainly James Milner, where they want to be a little bit more secure, a little bit safer.
“Curtis Jones is always looking forward and always willing to make that little pass forward, so that will stand him in good stead because I think he is a little bit different to the rest of the players we have.“
Jones has scored three goals and provided one assist across all competitions for Liverpool so far this season and will be eager to add to the tally.