Kane struck six times in Russia as England reached the semi-finals, although that figure included three penalties and a fortuitous backheel.
Pochettino congratulated his star striker when he dropped in on Spurs’ training ground during his holidays, only for the Spurs striker to respond: "No gaffer. I'm really disappointed, I should have scored more goals."
A surprised Pochettino told MailOnline Sport: “He said scoring six goals and reaching the semi-finals was not enough. That is why I love him.
“I love his energy and desire. For him it is not enough to finish fourth in the World Cup or second or third in the Premier League.”
Tottenham had the most players from any club in the World Cup last four – an impressive feat but also a hindrance ahead of the new campaign, with the nine-strong contingent only returning to training on Monday.
Spurs visit Newcastle in the Premier League opener on Saturday, with Pochettino unlikely to risk those who went deep in Russia despite failing to add any players in the summer window.