The Gunners are in a battle with Manchester City to land Rice, 24, with The Athletic now reporting their bid is structured as £100 million with £5 million in add-ons.
Should West Ham accept the offer, it would set a new transfer record for a British player.
Arsenal previously put forward a £90 million package for the England international standing at £75 million plus £15 million in add-ons which was turned down by the Hammers.
City, meanwhile, made a formal offer of £80 million with £10 million in add-ons but were also met with a swift rejection.
Rice has one more year remaining on his current West Ham deal, but all signs point towards an exit this summer after chairman David Sullivan said he is expecting a departure.
He has made 245 appearances for the East London club and captained the side to European glory in the Europa Conference League.
