Martinez led Everton to a Premier League record tally of 72 points in his first season in charge at Goodison Park last term.
They were still pipped for UEFA Champions League football – ultimately the club's aim – but Osman says the mood is positive as the club prepare for another campaign.
"Can we improve on last season? I think so. With the group of players that we've got and the squad we've got. Absolutely. It's possible." Osman is quoted as saying in the Liverpool Echo.
"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.
"You've always get 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."
Much was made of Martinez's impact on Merseyside last term and Osman believes the Spaniard's spirit is infectious.
And the 33-year-old - who has been awarded a testimonial against Porto next month - is already excited for Everton's UEFA Europa League campaign.
"You can't help but feel his (Martinez's) positivity, you can't help but feel it and be sucked in by it and use it yourself to operate daily," Osman added.
"Everything he does is positive and when he talks to you and about you it's always positive.
"European nights are different and I'm glad the current players and young fans who didn't quite get to enjoy it last time will see how special the European adventure is.
"They are incredible nights – home and away. Away from home we had some incredible nights and at home, under the Goodison lights, the games are really special."
