Arsenal submitted a third offer for Rice worth £100 million ($127m) plus £5m ($6.3m) in add-ons that was accepted by West Ham on Wednesday, sources told ESPN. While there remain minor details to finalise, they are said to be a formality at this stage.
City decided not match or better Arsenal's offer Tuesday, sources said, and they are no longer in the hunt for the England international.
The reigning Premier League champions had a valuation in mind for Rice, 24, and believe Arsenal's bid has already exceeded it. City had an offer of £80m plus £10m in add-ons rejected on Monday.
The North London club have agreed personal terms with Rice, sources added. Also on Wednesday, Arsenal confirmed the signing of Kai Havertz from Chelsea for £67.5 million ($85.4m).
West Ham have long expected Rice, who has one year left on his contract plus the option of another year, to leave this summer after guiding the club to the Europa Conference League title earlier this month.
Arsenal had their opening two offers for Rice rejected, sources told ESPN, including a bid last week of £75m plus £15m in add-ons.
Manchester United have also been linked to Rice, but they have not been able to match West Ham's valuation.
City have already bolstered their midfield this summer with the arrival of Mateo Kovacic from Chelsea in a deal worth £25m plus another £5m in add-ons. Their interest in Rice was stepped up after Ilkay Gundogan's decision to join Barcelona on a free transfer.
Sources told ESPN that City are comfortable with their position on Rice having stuck to their valuations on players such as Marc Cucurella, Harry Maguire, Jorginho and Kalidou Koulibaly in the past. City missed out on all four players and believe their decisions have since been vindicated.