City’s offer was thought to have included an £80m initial payment but is short of the £100m the Hammers want, according to Reuters.
Arsenal are expected to make a third bid for the 24-year-old England midfielder.
The Gunners’ second proposal of an initial £75m in four payments, plus £15m in add-ons, was turned down by West Ham last week.
Rice’s contract with the Hammers expires in 2024, although it has the option for a one-year extension until the summer of 2025.
Following the club’s Europa Conference League victory over Fiorentina earlier this month, chairman David Sullivan said he was “99%” sure Rice would leave West Ham this summer.
Although they have lined up Chelsea’s Mateo Kovacic in a £25m deal, the loss of Ilkay Gundogan to Barcelona means City manager Pep Guardiola is keen to bolster his midfield options, with Rice viewed as the ideal candidate.