I don't get the notion that Arsenal are worried about a fans backlash, when they've just given Wenger a 2 year contract. Don't underestimate the amount of pressure they were under to change manager.
Plus, it's quite typical of football fans to be ready to riot after losing their star man, and then when the club replace him everything is ok again. Remember Newcastle selling Cole?
Once they get a shiny new toy, the faux outrage will soon die down. For Arsenal, the risk is prevent a player going to City for £50m or he stays for a year and goes to City for free. The business side of it just doesn't add up and that will always come first, especially to Arsenal. Most Arsenal fans, given that option, would say take the money and reinvest it now, as much as they will die inside seeing him in a City shirt. I'm sure they'd love him to go to Bayern but if the player doesn't want to go, they can't force him.
Granted, Arsenal will need to replace Sanchez with a high profile player to shut the fans up. Personally, i'd like to see them get Mpabbe. It gets us Sanchez without a doubt and gives the kid a few years to learn the league for us to go and buy him then instead :)