Gatekeepers News reports that the two clubs have continued alks are on how the deal will be structured for the 24-year-old.
It came after the third bid from Arsenal, having previously had two bids rejected after falling short of the £100m West Ham want.
Manchester City had earlier offered £90m for Rice, but this was rejected on Tuesday. The Citizens subsequently withdrew from the running.
Hammers captain Rice has been at the club since joining their academy from Chelsea in 2014.
Earlier in June, he led them to Europa Conference League victory, their first major trophy in 43 years.
He has scored 15 goals in 245 games for West Ham, having made his debut in the 2016-17 season after coming through the academy.
Rice has been capped 43 times by England, playing in all five of their matches at last winter’s World Cup.
His contract at London Stadium was due to expire in 2024.
The Gunner’s bid is comprised of an initial £100m plus £5m in add-ons. The initial fee equals the British record Manchester City paid for midfielder Jack Grealish.
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