The Nerazzurri coach is determined to mount a bid for the Serie A title next season after taking over at San Siro following a turbulent campaign under Andrea Stramaccioni
The Nerazzurri finished a disappointing ninth place last season, leading to the sacking of former boss Andrea Stramaccioni.
The 51-year-old is determined to improve on last season's performance.
"I guess you could say that Inter dreams of overtaking all of them. I didn't sign to come third," he told La Gazzetta dello Sport.
"At the start there are teams who are better. Juve won the championship, Napoli finished second, AC Milan third and Fiorentina fourth. But we'll play with everyone, this should be clear.
"We'll be the first to play in the Coppa Italia on August 18 and seeing as though we are not competing in the Champions League, we will aim to win it.
"I've won by beating Juve and I know what it takes. Inter also know [what it takes] as winning the Coppa Italia a few years ago helped the club to start to win."
The San Siro outfit have a number of veterans in their squad but Mazzarri has pledged to pick his team based on performances as opposed to age.
"I look at the field, not the identity card; who runs and plays well. But Inter has a good mix between youth and experience."
Mazzari also took time to reflect on his successful spell at Napoli and believes a move abroad is on the cards.
"The fans, the players and Napoli remain in my heart. I'll be cheering for Napoli in the Champions League. I do not rule out [coaching abroad]. But first I want to win with Inter."