There has never been any suggestion that I know of to the effect that Kane asked to join City last summer. Supposedly, there was a chat between Kane and Levy during which, according to the Kane camp, Levy agreed to let Harry leave if Spurs were not competing for honours. But in that event, the agreement would have been equally dependent on another club making an acceptable offer.
Kane will be pissed off regardless. He wants to leave. He wants to win trophies. And Spurs aren't anywhere close to doing that currently. But he has always been hugely focused on personal achievements. He wants to take Alan Shearer's top all time goalscorer crown. He will want to ensure that he retains his England place and the captaincy for next summer's World Cup. He will not want to waste an entire season sulking or going on strike.
So if Spurs and City can't agree on a price and if Kane is therefore made to stay with Spurs for another season, I would expect him to continue to be the consummate professional that he has always been - no matter that he is pissed off.