The 22-year-old has been capped once by the Three Lions, with his sole appearance coming against Sweden in 2012.
"A few of the England boys have retired in the past couple of months. Glen Johnson is in his 30s now. I am not sure how much longer he will play international football for," The Mirror quotes him as saying.
"The position is there for the taking. It's up to us all to do the business. It won't be easy, of course, but I, as much as any of those boys, want to be in that England squad. I believe I am capable of it."
Jenkinson is expected to make his debut for West Ham in tonight's Premier League match against Hull City at the KC Stadium.