The newly crowned Europa Conference League winners will reluctantly allow Rice to leave this summer after he turned down a new deal but value him in excess of £100m.
Arsenal lead the race for his signature, but their opening offer falls short of that valuation. Sources have told ESPN that the bid comprised an initial £80m plus a further £10m in add-ons.
Sources added that Arsenal are expected to return with an improved offer for the midfielder.
City are assessing their options and sources suggest that West ham would be interested in a player-plus-cash deal which would see Kalvin Phillips move to east London but no such proposal has yet been made.
Arsenal have identified Rice as their top midfield target and are keen on securing a deal swiftly so the 24-year-old can join their preseason tour to the United States.
Speaking after West Ham’s Europa Conference final victory over Fiorentina last week, club chairman David Sullivan said Rice is set to leave this summer with the player having two years remaining on his existing contract.
“We promised him he could go; he set his heart on going,” Sullivan said. “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 [in lost wages]. And he wants to go. You can’t keep a player who doesn’t want to be there.”