The Merseyside club last claimed silverware in 1995, winning the FA Cup courtesy of Paul Rideout's goal in a 1-0 victory over Manchester United at Wembley.
However, expectation is increasingly high around Goodison Park after Everton followed a fine debut campaign under Roberto Martinez, which saw the club finish fifth in the Premier League, with significant close-season dealings.
In addition to signing Gareth Barry on a permanent deal and convincing star midfielder Ross Barkley to sign a new long-term contract, Everton announced the £28million signing of Romelu Lukaku from Chelsea on Wednesday, with the Belgium forward and Barry having impressed when on loan last season.
Osman, who is preparing for a testimonial against Porto on Sunday, therefore feels his side are well placed to contend for honours.
"I think we can make the next step," he told reporters at Claire House Children's Hospice, one of the charities that will benefit from the proceeds of his testimonial.
"We have got quality players and the manager has waited to get the right players, who are going to fit in on and off the pitch.
"He knows what extra we need for the demands we've got this season with the Europa League. He's a very clever manager and I think we will have the squad to compete for every competition we are in.
"Silverware isn't the next target for me, it has always been the target since I got in the first team.
"It hasn't been achieved yet but it is something that seriously needs to be addressed with the quality players we have got in our squad and have had over the past 10-12 years. It's about time we turned that quality into a trophy."
