Bale scored both goals at Carrow Road as Spurs moved to within seven points of the Barclays Premier League summit with an easy 2-0 win over Paul Lambert's team.
The Welshman opened the scoring after turning home Emmanuel Adebayor's pass from 12 yards in the second half, before lobbing John Ruddy with an exquisite chip after racing through on a dash that began on the halfway line.
Barcelona have watched the 22-year-old this season and the player himself has admitted that he would be interested in a move to the continent in the future.
But after spending all summer fending off interest in Luka Modric, Tottenham are determined to hang on to their best players, and for that reason Redknapp insists Bale is going nowhere.
"It would only be a Barcelona, a Man City or Real Madrid who would be able to pay for him," Redknapp said.
"They are the only ones who could afford to pay the transfer fee and his wages, but he isn't for sale. The chairman has said he isn't for sale.
"If we sell him we've had it haven't we? If you looking to go somewhere you don't go and sell a player like him."
Bale excelled in a more central role given to him on Tuesday night, where he played as the left-sided prong in a three-man attack that also included Rafael van der Vaart and targetman Emmanuel Adebayor.
Redknapp was delighted that the move came off, adding: "I played him and Rafa up there, rather than sticking him out on left wing because when he picks it up and runs at you through the middle and he pops up in positions between midfield and defence, it's difficult to pick him up. When he finds those holes he is unplayable."
Norwich boss Lambert, whose team rarely threatened Spurs 'keeper Brad Friedel, was highly impressed with Bale, but refused to criticise his men for failing to register another clean sheet for the 18th time this season.
The Scot said: "Bale is top class. He did ever so well. I was disappointed with the goals we conceded but you have to remember who we were playing against.
"We now have got Fulham which is also a tough game but hopefully we can give a good account of ourselves."