Brazil edged past Chile in the second round after a dramatic penalty shootout, while Colombia coasted to a 2-0 victory over Uruguay to book their place in the last eight.
Colombia have won all four of their games so far in this World Cup and are now being tipped as dark horses to go all the way.
Colombia have beaten Brazil only twice - the last time in the 1991 Copa America - but the teams have drawn the last four matches they played.
Brazil, who are aiminig for their sixth World Cup title, have reached the quarter-finals in six consecutive tournaments, but have been eliminated at this stage in the last two.
The hosts will be boosted by the news that star man Neymar has shaken off knee and thigh injuries to take his place in the side.
However, Brazil will be without key defensive midfielder Luiz Gustavo as he is banned after picking up his second booking of the tournament against Chile.
Tottenham midfielder Paulinho is likely to replace Gustavo in midfield alongside Manchester City's Fernandinho.
Luiz Felipe Scolari must decide whether to stick with under-fire striker Fred, who has struggled up front so far in the tournament.
Scolari also hit back at suggestions that his Brazil side have had problems with the expectations on them to win the World Cup on home soil.
"It's normal to have people demanding that we play better," said Scolari. "But it's also normal what we have been seeing in this World Cup so far. There were a lot of even matches.
"There is no difference between teams that have tradition and world titles and the rest of the teams. Matches are being decided on penalties, in the final minutes, on mistakes."
Colombia have no fresh problems to deal with as they prepare for their first-ever World Cup quarter-final.
Jose Pekerman is expected to keep faith with the side that overcame Uruguay, with Colombia boasting the second best attack at the tournament having scored 11 goals and only conceding two.
Pekerman admitted the game will be hard to call, with world-class players on both sides.
"To progress to the semi-finals is a high leap and this is a defining match because both teams will be at their peak of their performances," said Pekerman.
"On both sides, we can observe great players, and when you have great players on both sides of the pitch, the match will be interesting and very even."
Colombia wil again look to the tournament's leading scorer, James Rodriguez, to inspire them against Brazil as he looks to add to his tally of five goals.
Paul Merson’s prediction
Colombia have been the form team of the World Cup, but as with France I would ask who have they played? Let’s be honest, if you take Luis Suarez out of the Uruguay team they aren’t very good and they wouldn’t have beaten England without him (how many games do you see us play in international football with 60% of the possession??) Colombia put them to the sword, but I think they might come unstuck against Brazil at home.
I think Brazil have been absolutely shocking and if they win the World Cup then they will probably be the worst team to ever win it. Neymar is the stand-out player and if he doesn’t turn up to the party then you can’t imagine Hulk doing anything for them. Or Fred. Then you look at the back and the two full-backs – Marcelo and Dani Alves – can’t defend for toffee and we all know about David Luiz.
But the thing is, when you have got a record of playing 40-odd games at home without losing then you’re going to be hard to beat. I will go for Brazil because they’re at home and the Colombians might end up playing against the shirt, rather than the players.
PAUL PREDICTS: 2-1