Manuel Pellegrini's men sit third in the Premier League table and have already booked their place in the knockout stages of the Champions League.
The Blues also qualified for the semi-finals of the League Cup last week, while they are yet to begin their FA Cup campaign, and Hart admits that a club of City's stature must be aiming to win every piece of silverware.
"It will be a successful season if we win all the trophies," Hart told Sky Sports.
"We can't say we want to lose that one or we accept we are going to lose then. We have to go out to win all four and see what happens."
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Manchester City have struggled to make an impact in Europe in recent seasons, but Hart is optimistic about his side's chances this term.
"We are ambitious as a team, as a club and as individuals," he added. "We want to do well in the Champions League and I have always defended us from what people say, I don't care what people say about Manchester City.
"We know where we are, we've been here a long time and we know what we are trying to achieve. We want to do well for the club and we want to do well for each other."
City won the Premier League and League Cup in 2013-14, but have never won more than two major trophies in one season.