The 69-year-old left the Hammers in 2016 after 43 years of service but told the We Are West Ham podcast he hopes to see his last superstar graduate repay the faith the East Londoners have shown in him for at least another season.
Asked about Rice's future, Carr said: "I'm a little bit concerned. I hope he stays with West Ham I really do.
"He's developing into a top player. His goal the other night against Watford - terrific.
"You could see the look on his face, the pleasure it gave him. And the hug he gave to David Moyes on the touchline when the goal went in sends great signs and signals he's happy there and wants to stay.
“Unfortunately, sometimes agents get involved – I'm not saying this is Declan's agent - but if someone says 'we're going to double your money if you come to us', it's hard to turn down and let loyalty be the deciding factor.
"I hope it doesn't come to that. I hope he just comes out and says 'I'm happy to stay at West Ham, I'm still developing as a player, they've brought me from obscurity, released from Chelsea, and now into the England team'."