The thing is Sullivan was never going to accept the 1st bid - whatever bid Arsenal make he will go to the other clubs and ask them to beat it so it doesn't make sense to go all in. It does seem Rice prefers Arsenal (I guess for family reasons plus Arteta has been courting him for a while) and if he went to City he has the formidable presence of Rodri to get past in order to get into the team (not to mention De Bruyne, Gundogan etc) whereas he'd be a guaranteed starter at Arsenal. Plus maybe he sees it as more of a challenge at Arsenal (although the pull of working under Guardiola and surrounded by amazing players must be a big draw too).
Seems like Arsenal have gone back in this evening with a 2nd bid of £100m.