Inter booked their Champions League return on the final day of last season but have made a strong start to the 2018-19 campaign.
Defending champions Juventus, though, are nine points clear of third-placed Inter having only dropped two points so far this season.
Inter announced the appointment of 27-year-old Steven Zhang, the president of the club's owners Suning International, as their new president this week.
And Spalletti says that talks with Zhang have clarified Inter's goals for the campaign, with a place in the top four their primary target.
"Inter as a club are only satisfied by big results in big matches, taking up a place in the top four," Spalletti told a news conference ahead of Monday's trip to Lazio.
"To stay in the Champions League is now a goal, it's like winning a title, and there are strong teams at the top as there have been for years.
"The team has put in excellent performances recently and picked up excellent results."
Spalletti was unhappy with what he perceived to be a passive approach from Inter in their 2-0 Champions League defeat away to Barcelona.
And the head coach promised there will not be a repeat when Inter take on Lazio at the Stadio Olimpico, with centre-back Stefan de Vrij set to face his old club.
"We’re looking to take everything that we have available to us into account. What is vital is the sign that the team showed me yesterday [Saturday]," Spalletti added.
"They showed that they won’t passively accept situations that don't go well, they'll react when things don't go as we would have wanted.
"De Vrij is an easy person to analyse, you just need to look at him. He's transparent, clear and honest. He's an incredibly enthusiastic person. He trains brilliantly, he's a strong player and a very calm footballer, you can see that on his face. That's fine for us.
"From Lazio v Inter last season, I remember a fanbase in celebration after we reached an important goal [Champions League qualification] amidst a number of difficulties, especially considering the strength of the opponent.
"We can only look to do it again. We don't want to stop after such wins but we want to do it again and win more big games. We're going to Rome to play."