Unlike in previous years, the national team head to Poland and Ukraine as outsiders to win the tournament, and Jagielka is hoping they can now surprise people.
“I’m sure if we win a game we’ll be favourites and if we lose a game we’ll be a waste of space,” he told reporters.
“It’s a hard one to call. It would be better for the squad if there’s a little bit less expectation on us. Hopefully we can go under the radar until the latter stages of the tournament and hopefully get further than we have.
“If we’re being realistic we haven’t won anything for a number of years and maybe that’s because we don’t deserve to so hopefully this time round we can perform better and see where that takes us.”
The Everton centre-back also suggested that not much has changed since England’s friendly clash with France in November 2010 - ahead of their meeting in the tournament opener on June 11.
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He added: “I’m sure their squad will be quite similar to what it was back then and ours isn’t too different. It’s going to be a slightly different atmosphere and a different sort of onus on the game.
“Obviously it’s a tournament game, not a friendly so we’ll need the points to top the table and hopefully go through.
“We’ll be going out there to win the game but I think if we take a draw and get two other good results in the group then we’ll take that just as much.”
Jagielka went on to reveal he has enjoyed training under new boss Roy Hodgson so far, adding: “Its been good. We’ve obviously not had too long with it but a 1-0 win [against Norway] is a decent start.
“We didn’t play amazing but we’re still getting used to the tactics and the different roles that some of us have to play. We’ve got to look upwards and we’re going into a tournament and hopefully we can do well with him in charge.”
The 29-year-old also expressed excitement at the prospect of playing for his country alongside his former Toffees team-mate Joleon Lescott.
He continued: “Yeah definitely me and Joleon have known each other for about 10 years now so that’s obviously useful for England.
“We’ve played together a lot at Everton him got himself a fantastic move to Manchester City and has won himself a Premier League medal this season so his career has gone onto fantastic things and I think I’ve done ok for myself.
“It’s always nice to have a familiar face next to you, especially at international football where the standards are so high. I think the last three games with had we’ve had three wins.”