The Spanish champions host the first leg on February 13 with the return at Old Trafford on March 5 and, with the defence of their league title faltering, success in Europe is taking on even greater significance.
"I think there's more pressure at Madrid than at United," the Portugal forward, who was named World Player of the Year as a United player in 2008, told www.fifa.com.
"I've been here for three-and-a-half-years now and I think that's the case because everyone involved is totally focused on winning the club's 10th European Cup.
"That creates a certain amount of anxiety, which can be felt at all times.
"We haven't started the championship very well for sure. We know that La Liga's an uphill struggle for us now, but nothing's impossible in football. There's the Champions League and the King's Cup too. There's plenty for us to win this season."
A 0-0 draw at lowly Osasuna on Saturday left Jose Mourinho's side 18 points adrift of unbeaten La Liga leaders Barcelona in third place.
Their biggest deficit on their arch-rivals in a 20-team league was in the 1987/88 season when Real trailed Barca by 21 points at the midway point of that campaign.
Before then, however, Real also have a King's Cup quarter-final against Valencia which starts on Tuesday.
"I think that, yes, the pressure is too great and that sometimes it can affect us," Ronaldo added. "But we have to try and rise above it and not let what people say get to us.
"Every Madridista wants that 10th European Cup - we're more than aware of that. Last season we fell just short, which was very frustrating.
"Fortunately we've got the opportunity to set the record straight this year, though the fact we face Manchester United next makes it more difficult. It's not going to be easy and I reckon the odds are about 50/50, but I'm still very confident."
Mourinho has been heavily criticised in the local media and whistled by home fans after their recent poor run of form.
"People do protest and you have to respect that, though I think they should be a bit patient with him," Ronaldo said.
"To me it's very clear that Mourinho always wants what's best for the club and will fight to the death to defend it.
"In my view he's the best coach in the world, he's got loads of experience and has won everything there is to win."
The 27-year-old's contract with Real runs until 2015 and some media reports have linked him with possible interest from big-spending Paris Saint-Germain. But Ronaldo was adamant he would complete his current deal.
"I want to see out my contract at Real Madrid," he said. "I'm very clear about that. After that, well, I don't know what'll happen in the future."
