Real Madrid coach Zinedine Zidane hailed Kane as the "complete player" ahead of the clubs' first Champions League tie in October while Manchester United have also been credited with an interest in the 24-year-old.
Kane signed a new five-and-a-half-year contract in 2016 worth more than £100,000 (113,000 euros, $135,000) a week but his wages are still modest in comparison with the world's leading players.
After scoring twice in Tottenham's 3-0 FA Cup third-round win over Wimbledon on Sunday, Kane was asked what the club need to do to secure his long-term future.
"I've always said, just keep progressing, keep getting better," Kane said. "We want to start winning trophies so that's the aim. As long as the club keeps doing that then I'm happy here."
Pochettino said Philippe Coutinho's move from Liverpool to Barcelona shows there can be no room for complacency, adding the situation would need to be managed carefully.
"Harry is so special," said Pochettino. "Because he loves Tottenham and he has always been at Tottenham.
"But we need to be clever how we manage him. The player needs to choose to stay here, you cannot force the player to stay here. Our job is to try to work together and try to achieve everything we want at the club and of course Harry is such a special player. I told him only a few special players can spend all their career in one club."
Kane was asked about players now having the power to dictate their own futures.
"If a player wants to go then why would you stop him?" Kane said. "He's not going to be in the club, he's not going to want to play every game, he's not going to put his heart on the line."
Tottenham are reportedly preparing a new deal for Kane worth £200,000 a week but the striker was not aware of any planned negotiations. Pochettino said the Coutinho transfer showed how difficult it was even for a club the size of Liverpool to keep hold of their best players if they wanted to leave.
"With Coutinho, I think Liverpool wanted to keep him but it is difficult to keep him, and look what happened with Cristiano Ronaldo with Manchester (United), Zidane with Juventus, (Luis) Figo with Barcelona.
"It is a lot of examples that (show) it is so, so, so tough for everyone to keep your best player if the player in some moment turns their mind and says 'now I want to leave'.
"That is why it is so important how you care (for) your players. That is why, how you need to anticipate things."...