Right-back Jenkinson, a £1million capture from Charlton last summer, also says the Gunners squad is aware they must try to collect some silverware in the coming campaign.
He told the Daily Star Sunday: "This season will be huge. We feel it as much as the fans and we desperately want to win a trophy and be successful.
"That is something we are going to work on as a team this season and we will deliver for the fans, hopefully."
Jenkinson, whose dad has had a season ticket at Arsenal for more than 40 years, spent three months out injured with a bad back last term - but is confident he can continue to improve and challenge Bacary Sagna for the starting right-back spot.
He said: "The more minutes I play, the more comfortable I am to push on. I can't wait for the new season."