The right-back admits he did not harbour such high expectations at the start of the campaign but has had belief instilled in him by West Ham's recent form.
Sam Allardyce's side are currently sixth in the Premier League, just one point off fourth.
And Jenkinson believes there is no reason the Hammers should not aim for as high a finish as possible - which could mean Europa League football, or more, next term.
“With the players we have got this season we want to be aiming for a European spot," the 22-year-old told Sky Sports News.
"I think that’s something we can achieve, maybe at the start of the season I didn’t think we would be as high as we are but it’s fantastic that we are and we have surprised ourselves and a lot of people.
"We deserved more out of the game on the weekend, that was frustrating, but the performance levels haven’t really dipped to be honest. As long as we keep the performances high, the results will come."