City have decided not to match Arsenal’s latest offer worth £105million for Rice.
West Ham have not accepted the latest bid as they are unhappy with the payment structure of the proposed deal, and will have hoped City bid again.
But the Premier League champions will not be matching what Arsenal have offered, ending any rival interest in the midfield.
City have a history of not going over their valuations for players, or pursuing players not desperate to join, and did so last summer when Chelsea blew them out of the water to sign Marc Cucurella from Brighton for more than £60m.
Arsenal will now look to finalise a deal for Rice with West Ham ready to negotiate.
The Gunners have had two bids rejected for Rice so far, who wants to move to north London.