The England midfielder became only the third Hammers skipper to lift a trophy, after victory over Fiorentina in the Europa Conference League.
However, Sullivan admitted a gentleman’s agreement means they will allow a player who still has two years remaining on his contract 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.”
Rice said after the game that it was “not goodbye yet”.
He told BT Sport: “At the moment there’s a lot of speculation about my future.
“There is interest from other clubs, that’s out there, but ultimately I’ve still got two years left at West Ham.
“There hasn’t been nothing yet (offers). Let’s wait and see, who knows.
“I love every minute of it here, you can see the smile on my face.
“It’s not about that tonight, it’s about celebrating and that’s what I’m going to do now.”
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