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Premier League champions had offered £80m up front plus £10m in add-ons for the England midfielder but the bid is short of West Ham's valuation.
Arsenal are still pushing to sign Rice and remain in talks with the Hammers, who are expecting their London rivals to return with a third bid after having two previous offers rejected.
Arsenal had a club-record £90m bid - comprising a £75m fee and £15m in add-ons - turned down by their London rivals earlier this month after their opening offer thought to be worth £80m plus add-ons was also rejected.
Work now continues behind the scenes at the north London club over a new package. It is understood Arsenal are working to their own plan and any further bid will not be in response to what City are doing.
West Ham's valuation of Rice remains £120m, but it is understood the club would accept £100m plus a player.
Rice will leave West Ham in the close season after the midfielder captained the club to their first European trophy in 58 years.
Hammers chairman David Sullivan said a gentleman's agreement means Rice can leave despite a year remaining on his contract, with the option to extend it by a further 12 months.