The Toffees enjoyed a encouraging campaign under Roberto Martinez last term, and had fourth place within their grasp until the final few weeks of the season.
The Blues eventually finished fifth as Arsenal snapped up the final spot, but Osman feels that his side can go one better next term and qualify for Europe's premier competition.
Speaking to the Liverpool Echo, the 33-year-old said that the quality of the squad means Champions League qualification is a realistic target heading into the next season.
He said: "Can we qualify for the Champions League? Absolutely. It’s possible. You’ve always got to aim higher."
“We want to aim for the Champions League next season, we want to win all the cups we’re in. That’s what the aim is. I think with the group of players we’ve got it is achievable."
"With everything in football the second season is always the hardest. Because people know what to expect so it’s up to us to raise our game again. When you look in the eyes of the players around the dressing room I think we all think it’s achievable."
"If we went into a season thinking we wanted to try and do the same as what we did last season, or try and get close to what we did last season you’re not going to achieve anything."
If Everton are to qualify for the Champions League next term, they are likely to have to do so on a shoestring budget compared to their rivals.
The Blues have chronically been unable to compete financially with the Premier League heavyweights, and manager Roberto Martinez is expected to utilise the loan market once more this summer.
Gareth Barry's move from Manchester City was made permanent earlier this summer, while Martinez alongside owner Bill Kenwright, is still trying to engineer a deal for Chelsea striker Romelu Lukaku.