Kane, who has scored 26 goals in all competitions this season, received a maiden call-up to Roy Hodgson's England squad for the European qualifier against Lithuania on March 27 and friendly fixture with Italy.
English pundits have waxed lyrical about the 21-year-old this term, with suggestions even that Hodgson should "build" his team around Kane coming from another former Spurs man – Danny Murphy.
However, Arnesen - speaking at the launch the world's first Sporting Director Global Summit at the Etihad Stadium on April 29 - said: "With Kane it is fantastic that something can happen like this. To score 26 goals is a big surprise in his first year.
"But he has not played a [senior England] game yet.
"That is a bit of a problem in England. They put the English players under pressure.
"Ryan Giggs or Gareth Bale were a bit in the shadow playing for Wales, but [Paul] Gascoigne, [Wayne] Rooney and [David] Beckham were put under extreme pressure with people saying 'You guys have to give us the World Cup and the European Championship'.
"It is very difficult to live up to and it's not fair.
"Harry Kane has shown in eight months that he is the real deal, but to become a [saviour] figure?
"Look at Jack Wilshere. Two or three years ago he was number one. Come on, give them a break.
"You don't hear that in other countries. You go over the top. That is the problem with the British press. Take it easy.
"You can be a great player for Tottenham, but to be a great international player takes time."
Arnesen also claimed that Football Association chairman Greg Dyke's declared aim of England winning the World Cup in 2022 in Qatar was not fanciful.
"To go in and say we're not fooling around is fantastic," said Arnesen.
"England are the richest federation, the richest football nation in the world with all the TV money. If used wisely with the sporting tradition England has, it is not unrealistic.
"But you don't go from not being able to get to the semi-finals to be world champions.
"You need to come further than you are now in the European Championship before winning something. But I like the way he [Dyke] said this is my goal."
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