The winger, signed from Tottenham on deadline day, has high hopes for his new side.
The Blues endured a highly disappointing 2014/15 campaign as they struggled to deal with the added fixtures from the Europa League, finishing 11th in the standings in a season that fans will be keen to forget.
That came after they had narrowly missed out on Champions League football in Martinez’s first season at the club, and Lennon believes they can challenge for a similar position again this term.
"If you look at the quality the manager has brought in, the squad is getting so much stronger and I just can’t wait to help the team now,” he told evertonfc.com.
“Towards the end of last season, we got on a good run and this team’s definitely got the quality to be challenging up at the top end of the table. The season before last they finished fifth and I don’t see why we can’t be up there again with this team and with the quality it has.”
Can Everton challenge for a top four place?
While there is no denying the quality of Everton’s starting XI, it may be their bench that lets them down in the hunt for a top four spot.
They have largely failed to add to the squad that challenged Arsenal two years ago, although new recruits such as Lennon, Ramiro Funes Mori and Tom Cleverley will be hoping to raise the collective standard.
The problem is - as is always the case - the competition the Toffees face to reach the Champions League is fierce, with Manchester United and Liverpool strengthening heavily this summer and more likely candidates emerging from the chasing pack.
Tottenham Hotspur have also added wisely and will fancy their chances, meaning the the scrap for places will be keenly contested, and a Europa League spot may be a more realistic target for Martinez's men this term.