The Blues remain in the hunt for a historic quadruple in 2019.
A successful defence of the Carabao Cup has already been made, while the current title holders are back on top of the Premier League table.
They are also through to the quarter-finals of the Champions League, where they will face domestic rivals Tottenham, and Mills says that the FA Cup sits third on their list of priorities.
“If City were to win the FA Cup and the League Cup and nothing else, it would be considered a failure with the squad they've got and how good they were last season.
“That sounds ridiculous; you win two trophies and it's been a poor season.
“But I think the fans would go with the league, they want to retain that - they still haven't fallen in love with the Champions League because of the issues with Financial Fair Play and the amount of players you are allowed in the squad.
“I think the owners want the Champions League, that's what they came in to do - it's the toss of a coin, they'd probably take one or the other.
“There's a huge split between everybody, fans, owners, players, manager - and Pep's not won it with another club so he might want it.”
City reached the semi-finals of the FA Cup in Guardiola’s first season at the helm, before falling at the fifth round stage in 2017-18.
The Blues last lifted the trophy back in 2011 when a solitary strike from Yaya Toure edged them past Stoke at Wembley and secured a first piece of silverware in the Sheikh Mansour era.