The Gunners saw their initial £80m offer rejected last week but soon made an improved proposal worth an initial £75m plus £15m in add-ons.
The Hammers have also knocked back the offer as they are holding out for a minimum of £100m, as per the Guardian.
Despite seeing their second bid rejected, Arsenal remain hopeful of signing the 24-year-old this summer.
However, Manchester City could also enter the race for the highly-rated midfielder.
Ilkay Gundogan’s future at the Etihad Stadium is uncertain, with the 32-year-old’s contract expiring this summer.
City want to extend his deal, but the German is yet to decide his future.
If he leaves, Man City will likely enter the transfer race for Rice.
Manchester United have also been credited with an interest in the Irons star, but they have yet to make a concrete offer.
The Red Devils are also unlikely to make a move before the new owners take over the club from the Glazers.
Rice is one of the best defensive midfielders in the world right now. He has been a consistent performer for the Hammers since breaking into the first team in 2017.
The England international has made 245 appearances in all competitions, bagging 15 goals.
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