Manager Roberto Martinez sealed Europa League football for the Merseyside outfit after his team finished fifth during the Spaniard's first season in charge.
The Toffees finished seven points behind fourth-placed Arsenal, but Osman believes that there is no reason why he and his teammates cannot challenge again.
"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," the Liverpool Echo quotes Osman as saying. "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."