Fact City have yet to bid confirms the theory Arsenal are Rice's first choice. I think City are monitoring and if for whatever reason Arsenal cant do the deal, City will prevent themselves as an alternative. Or your interest was passive at best, and West Ham blew it up to hurry Arsenal into action.
I actually think Arsenal and West Ham are talking behind the scenes now that a 'bidding war' is out of the question. It was in West Hams interest to receive and review bids, and not reveal their hand as to encourage a 'bidding war'. However, Rice seems resolute in his desire to move to Arsenal so any notion of a 'bidding war' is dead in the water. West Ham know if they are getting 100m for Rice, they are getting it from Arsenal. So as of right now, I think both clubs are in talks about the deal to bring it to a conclusion. I dont think Arsenal will prepare a bid and sent it to West Ham to review, both clubs are negotiating terms as we speak. The next major development will not be an Arsenal bid. Rather it will be an agreement between West Ham and Arsenal.
Major coup for us if we do it. Also lays down a marker for any potential suitors in Odegaard or Saka. If we can bid 100m for Rice. We can reject 100m for Saka or Odegaard.