Andre Villas-Boas' side ended the season with a win over Sunderland, but despite setting a new club record for points won in a Premier League season (72), they finished fifth and missed out on Champions League football.
PFA and FWA Player of the Year Bale scored another long-range screamer to seal the win and speculation is now rife that he will leave White Hart Lane this summer.
But Redknapp, son of former Spurs boss Harry, says the club must end their policy of selling their star players each summer and build a team around Bale that is capable of challenging for honours.
He said: "Two years ago I remember [former Manchester City assistant boss] David Platt called my dad and asked about buying Luka Modric and Gareth Bale. It's going to be exactly the same again.
"I've played for that club and I've seen it year in and year out that they sell their best players. Michael Carrick had a great season and they decide to sell him. Dimitar Berbatov? They sell him. Luka Modric? They sell him.
"Why don't they keep hold of their best players? Why don't they change the structure and go and give Gareth Bale what he would get if he was to go somewhere bigger.
"He might want Champions League football, but he also might want to be a big fish in a small pond. I've met Gareth plenty of times and he doesn't strike me as that sort of a boy.
"I just feel that he can still develop there and get even better and Tottenham can build a team that can challenge for the Premier League.
"There's no reason at all why they can't. They've got the money. Let's not kid ourselves, Joe Lewis and Daniel Levy who run the club have got all the financial backing to keep Gareth Bale.
"There's only reason if they don't want to do it and that's if they don't feel it's worth it and they don't want to push on.
"Tottenham can now push on, there's no two ways about it."
Bale equalled the goalscoring record for a Tottenham player in a single Premier League season and his 21 strikes earned his team an incredible 24 points this year.
And Redknapp says the Welshman is integral to the club's future if they are to become a force in English football.
"They have to keep Bale," he said.
"He is so important to that football club. He is now the figurehead, he is the crown jewel of the Premier League.
"What he has done this season, scoring nine goals from outside the box - the most in Europe - you have to build a team around him.
"If you let him go it would be a crime."