The 22-year-old has scored 42 Premier League goals for Tottenham since the start of 2014-15 and boosted his hopes of a starting role with England at the tournament in France by netting in their 3-2 win over world champions Germany in Berlin.
Kane understands that his stock is rising in the game and feels having to cope with closer attention from opponents is something that has to be overcome if he wants to reach his goal of becoming one of the world's top stars.
"Maybe I will be a marked man in the summer," he told reporters. "Again, that is football.
"If you want to be one of the top players in the world, you have to deal with that. I am happy for the challenge.
"I want to keep working hard, to keep getting better and better and to have more nights like that on Saturday, scoring more goals for England because they are special. You dream about scoring against teams like Germany on the big stage."
Kane did not want to be drawn on whether manager Roy Hodgson should pick him over captain and record goalscorer Wayne Rooney when the Manchester United forward returns from a knee injury, but revealed his confidence was sky-high.
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"That is not my decision," continued Kane. "That is down to the boss. Wayne is a fantastic player, he is our skipper and Roy has made it clear he wants him involved.
"All I can do is keep working hard and give the manager a problem, but I feel like I can score against any team in the world. You have to believe in yourself as a striker, that if you get the opportunities you will put them away."