The Gunners saw an initial offer worth £80million rejected last week and on Tuesday a second, worth £75m plus £15m in add-ons, also dismissed by the Hammers.
Rice is Arsenal’s priority summer signing and are expected to return with an improved offer as they look to secure the midfielder’s signature quickly.
West Ham have set an asking price of £100m for their captain and are prepared to stand firm with Manchester City weighing up a rival offer.
Rice has rejected a £200,000-a-week offer to stay in east London and with two years left on his contract, the club know this is the last chance to cash in big on their skipper.
LIVE BLOG: All the latest transfer news and rumours
Arsenal have also been linked with Romeo Lavia, retain interest in Brighton’s Moises Caicedo, and hope to land Chelsea forward Kai Havertz for around £60m despite a £70m asking price.
