You need the right ingredients for a rivalry. Liverpool and City fans don't really give two shits about each other, there's no history etc, managers are generally respectful. One season of competitiveness isn't going to change that.
You have the ones with history like United/Liverpool, Arsenal/Spurs and to a lesser extent, Arsenal/United, but there's generally not been enough riding on those games recently, so the games themselves will lack that natural spark.