A 2-1 victory over Fiorentina in Wednesday night’s final saw him become only the third Hammers skipper to lift a major trophy, following Bobby Moore and Billy Bonds.
However, Sullivan admits a gentleman’s agreement means they will allow a player who still has two years remaining on his contract – one year plus an option – 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 fans have been serenading Rice with the song ‘one more year’ for the last few months.
Yet the 24-year-old remains tight-lipped on his future, and said: “I’ve still got two years left. I know it’s a generic answer but there’s so much speculation, I see it all the time, but I can’t help that, it comes with playing football every day and doing what I do.
“My focus, and it has been since the World Cup, was to win this trophy and keep West Ham in the Premier League. We’ve done that so I’m going to enjoy it.
“I genuinely don’t know what’s happening at the minute, that’s my honest answer.
“I’m completely blocked away from the whole situation because it would be totally wrong. I’ve still got two years left on my West Ham contract and until the day I go I’ll always respect that. That’s the main thing for me.
“The last two or three years it’s been non-stop about me – to read it myself is actually getting quite boring. Who knows what’s going to happen?
“It’s flattering to be wanted by a lot of clubs, but the badge on my chest is West Ham, I’ve got two years left on my contract and ultimately it’s up to the people above.
“Until the day I walk out of this club I’ll give absolutely everything and I’ll wear my heart on my sleeve for this badge.”
Sullivan also confirmed that there were no plans to dispense with the services of manager David Moyes after a rocky season ended on a high.
“We believed in him. We were under a lot of pressure from supporters, the press, everyone to make changes but we believe in honouring contracts,” he added.
“David has a contract with us until next summer and next summer we will look at it again.
“Until then, as long as he wants to remain manger, he will remain manager at West Ham.”