The Gunners had indicated they are willing to meet the Hammers' asking price of more than £100m but their initial bid is believed to be much lower.
That opening bid is believed to be £80m plus £10m add-ons.
Rice has his heart set on a move to the Gunners, although champions Manchester City are considering hijacking the deal.
Manchester United have also enquired about the player, however a move to north London still seems the most likely outcome.
The 24-year-old progressed through Chelsea's youth academy before making the move to West Ham at the age of 14.
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After making his senior debut on the final day of the 2016/17 season, Rice has kicked on and made 245 appearances for the club.
Following Mark Noble's retirement, the midfielder was handed the captain's armband and lifted the Europa Conference League trophy in what club owner David Sullivan expects to be his final game for the club.
Asked if it was his last match, Sullivan told talkSPORT: "I think it has to be. We promised him he could go. He set his heart on going.
"You can't ask for a man who has committed more to us this season.
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"In due course, he has to get on and we have to get a replacement - or several replacements.
"It's not something we want to happen. We offered him £200,000-a-week 18 months ago. He turned it down.
"It's cost him £10million to stay at West Ham in that time (in lost wages). And he wants to go.
"You can't keep a player who doesn't want to be there."