This revelation comes mere hours after the Hammers ended their long-standing trophy drought with a historic victory in the Europa Conference League final against Fiorentina.
Rice, an influential midfielder for both West Ham and the England national team, became just the third captain in the club's history to hoist a major trophy following the 2-1 triumph in Prague.
However, Sullivan has confirmed that a pre-existing understanding with Rice grants him permission to pursue other opportunities, with Arsenal being strongly linked as a potential destination.
When asked about Rice's future with the club in light of the trophy win, Sullivan revealed to 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 move on, and we have to find a suitable replacement."
Sullivan expressed disappointment at the turn of events, stating, "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."
Despite acknowledging the interest from several clubs, Sullivan emphasized that formal offers for Rice will not be entertained until West Ham's participation in ongoing competitions concludes, out of respect for the club.
He added, "I think the offers will start to come today. Three or four clubs have shown interest, but while we're still playing, you don't make offers for players out of respect for West Ham."
The victory in Prague marked a historic moment for West Ham, ending a 43-year wait for a major trophy.
Reflecting on the triumph, Rice expressed his astonishment, saying, "I'm still in shock. It's incredible. I love this club; they've made me one of their own."
Regarding the interest from other clubs, Rice remained open-minded, stating, "Let's wait and see. Who knows?"
It remains to be seen which club will secure the services of the highly sought-after midfielder, but Rice's departure will undoubtedly leave a significant void in the heart of West Ham's midfield.