United will be favourites to get past German side Schalke, while Real's clash with arch-rivals Barca pits arguably the two best sides in the world against each other.
Redknapp said: "Barcelona for me look the best team I've seen in about 25-30 years. I must say when I see Barcelona play, I think they are amazing."
Yet, the Tottenham boss added of Jose Mourinho's Madrid: "You'd be a fool to write one of his teams off. What he's done has been incredible.
"Last year, I went with Inter Milan to win the Champions League.
"I just had a feeling that he could do it.
"The final's at Wembley this year as well, so I wouldn't write Man United off."
Spurs, meanwhile, were left to reflect on an unforgettable first season in the competition, which saw them beat both Milan giants before coming up short against the nine-time European champions.
And Redknapp challenged his players to do it all over again by pipping Manchester City to fourth place in the Premier League.
He said: "We've got to be up there every year. We should be.
"It will be even harder this year than it was last year because of the improvement Man City have made.
"I still think we've got a big chance. We've got tough games.
"We've got Arsenal here next week, then we've got Chelsea, we've got to go to Man City.
"But they're great games to look forward to and it's good to be in there playing these games and having something to play for still."
Mourinho gambled on four of his five players who were on yellow cards last night and saw Ricardo Carvalho pick up a booking that ruled him out of the first leg against Barca.
Mourinho, who was punished earlier this season for ordering his players to get themselves cautioned deliberately to earn convenient suspensions, used his post-match press conference to complain about double-standards.
Barcelona appeared to use the same tactic in a Primera Liga game at Villarreal earlier this month to ensure two of their players would be available for the upcoming league match against Madrid.
Andres Iniesta also picked up a convenient yellow in Barca's Champions League quarter-final first-leg win against Shakhtar Donetsk.
Mourinho said: "To be honest, as I always am, when we scored the fourth goal against Tottenham at the Bernabeu, my assistant told me, 'Yellow card to Cristiano (Ronaldo) and to (Ricardo) Carvalho'.
"And I told him, 'No way, because, if we do it, I am suspended for the semi-finals and my players are suspended for the semi-finals'.
"The next day, we are at home. Another coach did it, another player did it."