The latest offer was £75m up front, with £15m in add-ons but West Ham did not see those as achievable.
Arsenal will go back in with another bid and the feeling is they are getting closer as Mikel Arteta wants business done and his squad set before the summer tour.
There is the possibility that Manchester City come in but their business is currently dependent on what players leave, allowing Arsenal an opening, while Chelsea are thought to be monitoring the situation.
Should they be successful, Arteta could well oversee over £200m of expenditure, as the club aims to bring in Rice, Chelsea's Kai Havertz, Southampton's Romeo Lavia and Ajax's Jurrien Timber.
Some of that will be offset by sales, as Arsenal are prepared to listen to offers for a few players, including Kieran Tierney, Thomas Partey and then one of Eddie Nketiah and Folarin Balogun, with the Gunners eager to keep one as cover in attack.
Should Rice join the Gunners, the England midfielder would become the club’s record signing, beating the £72m Arsenal paid for Nicolas Pepe in 2019.
The 24-year-old lifted the Europa Conference League trophy in Prague on 8 June before owner David Sullivan said he believed that was the skipper’s last outing for the Hammers.
Arsenal finished second in the Premier League last season, with manager Mikel Arteta and football director Edu having identified the players they want to add to once again challenge Manchester City.
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