That is according to former left-back Scott Minto, who represented both clubs during a stellar 18-year professional career.
Rice began his career at Chelsea before being released at the age of 14 and joining West Ham, where has made 110 appearances across all competitions over the past four seasons.
West Ham manager David Moyes has insisted Rice is not for sale, but also admitted the club might not be able to stop Rice from leaving London Stadium.
Lampard wants to bolster his defence after his team conceded a whopping 54 goals in the Premier League last term.
Ben Chilwell and Thiago Silva are both expected to arrive this week, and reports claim Lampard wants a £70m deal for Rice to complete his summer shopping spree.
With the ability to play as either a centre-back or holding midfielder, Minto believes Rice would provide vital versatility for Lampard, as well as a leadership presence not seen at Chelsea since the days of John Terry.
"He could be the next West Ham captain for 10 to 15 years. The problem is that he is that good. Being a former West Ham player, I'd love to see him stay. Being a former Chelsea player, I'd love to see him go to Chelsea.
"And I can see him, as well as he has done in that holding midfield role, which I think he can do that for England, I wouldn't be surprised if Frank puts him in as a centre-back, because that's what he was initially. And Chelsea haven't really had a leader since John Terry. He's someone, I wouldn't want to say he could be the next John Terry, but I would say he could be a massive piece in Chelsea's jigsaw.
"And holding midfield in the way football is these days, not just in terms of the possession but also the quick counters, is a crucial position. So whether he plays as a holding midfielder for West Ham, whether he goes into someone like a Chelsea as a centre-half, his versatility is absolutely key. He would be an eight out of 10 in either position."
He added: "In terms of West Ham, if they could somehow keep him then that would be a massive statement. Every season he is getting better, better and better.
"I heard off someone I know that even Real Madrid were looking at him a year ago. Whether that is still the case or not I don't know. It might just be let's just keep tabs. He's just that good."