The North London club had an opening bid rejected last week and have now returned with a revised bid totaling around £90m ($115m).
Although the overall package is a similar amount, sources suggest more of the initial £75m ($95.5m) payment would be paid up front while the proposed £15m ($19.1m) in add-ons are seen as more attainable than previously indicated.
It remains to be seen if West Ham accept Arsenal’s bid or continue to hold out for a higher fee. Given they value Rice in excess of £100m, the expectation is it will be rejected.
manchester city hold an interest in Rice, but as yet, they have not chosen to enter the running with a formal offer.
West Ham chairman David Sullivan confirmed Rice, 24, would leave the club this summer, saying Rice “has his heart set on going.”