The Toffees finished fifth in the Premier League, with their total of 72 points beating anything achieved under Martinez's predecessor David Moyes.
They had to settle for Europa League football after missing out on what would have been the club’s second time in the Champions League, but Europe’s elite club competition is where the experienced 33-year-old says they must aim for.
"Can we improve on last season? I think so," said Osman. "With the group of players that we’ve got and the squad we've got it's possible.
"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 got to aim higher.
"We want to aim for the Champions League next season, we want to win all the cups we’re in. I think with the group of players we've got it is achievable."
Osman feels that the influence of Martinez, who moved to Goodison Park from Wigan following Moyes' ill-fated appointment at Manchester United, has brought the best out of his squad.
"You can't help but feel his positivity, you can't help but feel it and be sucked in by it and use it yourself to operate daily," added the twice-capped England international.
"Everything he does is positive and when he talks to you and about you it's always positive. You could have had a stinker for 10 weeks running but he will find a positive.
"As a player you can't help but appreciate that and want to play for a manager like that."