4 of the richest clubs in the world all in dire need of a midfielder like Rice could drive the price up. I don't think £120m is likely but close to £100m including addons seems quite possible if you, Man Utd, Liverpool and Chelsea all want him.
Keep in mind that some deals are structured in ways where a fee that seems insanely high isn't that bad once you break it down. A headline might say "Arsenal pay £100m for Rice", but if £20m of that fee is tied up in performance based addons like CL qualification, winning various trophies and stuff like that, it suddenly makes it a lot less egregious.
In other words, I highly doubt the people who say they wouldn't pay more than £80m for him would mind eventually paying £100m if the last chunk of money was paid as a result of winning silverware.
Not sure I agree with the dire need bit. I only think Chelsea really desperately need Rice for next season, and there are mixed signals whether he is a priority for then (even though he should be). That being said, I’m sure every club you mention would love to have Rice.
But yeah, I feel 80-100 would be a great deal for West Ham, and think it’s probably the upper limit for his profile.