Rice, 24, joined the youth ranks of the Hammers after being released from the Chelsea academy at the age of 14.
The England international has so far made 202 league appearances for the east London outfit since his debut for the senior team in 2017 and has scored nine goals.
He is considered a prized asset at the London Stadium and is widely expected to move on from West Ham for a substantial fee when the transfer window swings open for business.
Jarvis believes that Rice could give the Hammers a parting tribute by helping the club to win the Europa Conference League.
Terming the West Ham captain a legend of the club, he added that lifting the European trophy would be an incredible way to leave the London Stadium if he is prized away in the summer.
“He’s a West Ham legend”, Jarvis said on BBC Radio 5 live.
“He’s captain and he could lift this trophy and be remembered with the greats.
“If someone comes in for him and pays a lot of money, then it’s an incredible way to leave.
“But at the same time you can see his passion and desire for his club.
“Everyone obviously wants him to stay.”
Arsenal are currently considered to be the front-runners to secure Rice’s signature and it remains to be seen whether the Gunners will be able to capture the 24-year-old.
