The former Spurs midfielder believes the club have to hang on to Kane and their other best players as well as adding more quality.
Spurs were Leicester City's closest rivals for the Premier League title for the final months of the campaign, but ultimately suffered bitter disappointment.
Not only did their challenge to overhaul the Foxes falter, but a shocking 5-1 defeat to relegated Newcastle United on Sunday saw them slip behind arch-rivals Arsenal into third.
However, it has undoubtedly been a hugely successful season for Mauricio Pochettino's side, and much of that has been built on the goalscoring exploits of Kane, who followed up a phenomenal debut season last term with an even better domestic campaign, winning the Premier League Golden Boot with his 25 goals from his 38 games.
The 22-year-old striker has now established himself as one of Europe's stop strikers and is certain to attract interest from across the continent in the summer transfer market.
Redknapp, who played 49 times in two-and-a-half years for Spurs, says that the club must make sure not only that they rebuff any approach for Kane, but that they keep hold of the rest of their stars as well as adding more quality in the summer window.
The midfielder-turned-pundit told Sky Sports: "He's [Kane] a great example of hard work and dedication. All that is now paying off because a lot of young players don't do that anymore. He certainly has and that's why he has got to the very top.
"Tottenham now have to match his ambitions. For many years Tottenham have sold their best players and that's why they have not really kicked on. If they want to maintain this and become a force they have to keep those players and maybe add one or two."
In recent years, Spurs have built impressive sides but have lost their key players to top European teams, such as the departures of Luka Modric and Gareth Bale to Real Madrid in 2012 and 2013 respectively.
However, Redknapp is right to suggest that if they want to maintain and even build upon the standard set this season under Pochettino, they must hold their current side together and have all of their key stars still at the club come August.