The 26-year-old is in the provisional 26-man squad for the tournament in France, which starts on 10 June.
But he faces competition from fellow midfielders Fabian Delph, Ross Barkley, Jack Wilshere and Jordan Henderson.
"I'm just trying to do what I'm best at and if that doesn't get me in then I'll have to hold my hands up," he said.
"There's some real talent in the midfield, so if I miss out I'll make it a target [to get in the England squad in the future]."
Hodgson will announce his final squad ahead of the deadline on 31 May.
Drinkwater and his club team-mates took part in a Premier League champions parade through Leicester on Monday, with an estimated 240,000 people turning out to watch.
The former Manchester United trainee described the scenes as "unbelievable", but already has one eye on the Foxes' 2016-17 Premier League campaign.
He told BBC Radio Leicester: "We want to try to retain the trophy - it's a target, plus we're new to the Champions League at the club.
"Hopefully we can impress in that and retain the title at the same time."
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Leicester City midfielder Danny Drinkwater says he is not guaranteed a place in Roy Hodgson's final 23-man England squad for Euro 2016.