Spurs front man Kane is expected to play a key role for England at the international tournament in France.
Kane is likely to start in all of England’s matches, after establishing himself as their main striker in recent times.
However, Carragher insists that the 22-year-old cannot simply expect to score - as there are no guarantees that he will find the back of the net.
“Nobody should make such assumptions. Tournament football is completely different and who's to say how Kane, Alli and Vardy — the top scorer, the PFA Young Player of the Year and the Footballer of the Year — will do shouldering expectation in their first senior tournament,” he said.
“I think the world of Kane, he's a top player and the best English striker at present but he had a taste of tournament football 12 months ago with the Under 21s and — after scoring 31 times in his breakthrough campaign — failed to hit the target in three games. For that reason, he won't be taking anything for granted so why should anyone else?”
Kane goes into the European Championships in wonderful form, after netting 25 Premier League goals for Spurs last term.
The youngster has developed a reputation as one of the finest talents in Europe, and he will have the opportunity to showcase his skills in France.
England’s Euro 2016 campaign gets underway on Saturday, with their first match coming against Russia in Marseille.
England also have matches against Wales and Slovakia to play in the group stage of the tournament.