The England midfielder became only the third Hammers skipper to lift silverware after a 2-1 victory over Fiorentina in the Europa Conference League final on Wednesday.
However, Sullivan admitted a gentleman's agreement means they will allow the player, who has been strongly linked with a move to Arsenal, to leave.
Asked whether lifting the trophy in Prague was Rice's last action as a West Ham player, Sullivan told talkSPORT: "I think it has to be. We promised him he could go.
"He set his heart on going and in due course he has to get on and we have to get a replacement.
"It is not something we wanted to happen. We offered him £200,000 a week 18 months ago and he turned it down. You can't keep a player who doesn't want to be there.
"I think the offers will start to come today. Three or four clubs have shown interest but out of respect to West Ham, while we're still playing, you don't make offers for players."
West Ham's win in Prague ended a 43-year wait to win a major trophy.
"I'm still in shock," Rice said in the afterglow of the win. "It's incredible. I love this club, they've made me one of their own.
"There is interest from other clubs. Let's wait and see. Who knows?"
