City stepped up their pursuit of the England midfielder on Monday with an offer of £80 million ($102m) plus another £10m in add-ons. However, sources have told ESPN that West Ham have knocked back the bid and are holding out for at least £100m in guaranteed payments.
Arsenal had a bid of £75m plus £15m in add-ons rejected last week. Manchester United have also shown an interest in Rice, but there are doubts about whether they can get anywhere near West Ham's valuation.
The Premier League champions were keen for Gundogan to stay but were only willing to offer a two-year contract -- Barca offered a two-year deal with the option of a third. They were also reluctant to meet the Germany international's wage demand given he will turn 33 in October.
There remain doubts about the futures of Bernardo Silva, Riyad Mahrez and Kalvin Phillips. West Ham, according to sources, are one of the clubs interested in Phillips if he decides to leave the Etihad Stadium this summer.
West Ham expect Rice to leave this summer but have so far held firm with their demands to receive at least £100m. He has one year left on his contract at the London Stadium, but West Ham also hold an option to extend his deal by a further year.