The 21-year-old made his senior international debut as a second-half substitute back in November as a Zlatan Ibrahimovic-inspired Sweden ran out 4-2 winners.
Jenkinson, who represented Finland at youth level, was not selected in the recent England squad set to face San Marino and Montenegro but is keen to continue impressing at a domestic level.
"I want to get back into the England squad and I'm fighting to do that," Jenkinson told the club's official website.
"I've just got to be doing it week-in, week-out for Arsenal, I've got to be putting in good, consistent performances and hopefully that will catch the manager's eye. I'll be aiming to do that and hopefully if I do that a call-up will come my way."
Despite a defeat in his maiden England appearance, Jenkinson holds fond memories from the game at the Friends Arena - although Ibrahimovic's remarkable 35-yard overhead fourth goal does not rank highly.
"It was one those moments of magic where you're sort of blown away," Jenkinson continued. "It was an unbelievable bit of skill, the way he was running away from goal. He's pulled out a bit of magic and moments of class like that don't happen too often.
"Of course it was disappointing that he played like that on my debut, but sometimes in football when players do things like that, you just have to hold your hands up and say it was worthy of winning the game. But I made my debut and that was a fantastic moment for me as it is in anyone's career.
"As a kid you dream of doing that and it's almost beyond your dreams because you never actually believe you'll get there. That was a huge moment for me and of course for my family, just to pull on an England shirt for the first time.
"Hopefully it's the first of many, that's the way I like to look at it. If I'm doing well in an Arsenal shirt then England will happen in itself."