The 64-year-old's team have enjoyed a fairytale first season in Europe's elite club competition, topping their group, which contained defending champions Inter Milan, before knocking out their city counterparts AC Milan over two legs.
But, following Real's first home defeat of the season last weekend, Redknapp insists his Champions League debutants could inflict more pain on Jose Mourinho's side on Tuesday evening.
"We have a great chance. Of course we do," the Spurs boss told reporters. "We feel we can give anybody a game. Real Madrid are favourites but we come here very confident and optimistic that we can do the job.
"They have some fantastic players and we have to deal with that. But we have some good players who can cause them some problems too, and that's the way we'll approach the game."
Mourinho suffered his first home league defeat in nine years against Sporting Gijon on Saturday and Redknapp reckons all the pressure will be on the nine-time European champions to bounce back at the Santiago Bernabeu.
"I don't feel pressure. The only pressure [on Spurs] is that we play well. No-one expected us to be here at the start of the season and we have a big chance of winning."
And with the prospect of this year's final taking place at Wembley, the former Portsmouth boss said there was no reason why his side could not get past Real Madrid and make it to the final.
"We're not just coming here to turn up, we want to keep progressing. The dream is still there for us, we have to make that dream come true.
"We want to go all the way if we can. We know how difficult it's going to be but there is still a chance and we have to take it."
