Sources have told ESPN that Arsenal want to pursue a deal for Rice at the end of the season while Chelsea, Manchester City, Manchester United, Newcastle United and Liverpool are monitoring developments. Sources added that West Ham are expected to demand £100 million to let the midfielder leave the club.
When asked whether the victory was Rice's last match for West Ham, Sullivan told talkSPORT: "I think it has to be. We promised him he could go. He set his heart on going. You can't ask for a man who has committed more to us this season.
"In due course, he has to get on and we have to get a replacement, or several replacements. It's not something we want to happen. We offered him £200,000-a-week 18 months ago. He turned it down.
"It's cost him £10m to stay at West Ham in that time. And he wants to go. You can't keep a player who doesn't want to be there."
Sullivan's announcement came the day after Rice won his maiden European trophy as a player. He told BT Sport after the match that "he loves the club" and still has "two years left" despite ongoing speculation around his future.
Rice added: "My focus is playing for West Ham at the moment. I'm captain of this club and I can't speak highly enough of this place. Let's just see what happens. Let's wait and see. Who knows?"