Costa Rica international Campbell is on loan at Olympiakos from Arsenal and is relishing the opportunity to take on the Premier League champions, with the first leg taking place in Greece on Tuesday.
"We believe we can beat them," Campbell said in the Daily Mail. "Manchester United are a historic team, even as a boy growing up in Costa Rica it was a famous team, but we have a strong team."
United were one of the teams linked with a move for Campbell in 2011, but he had little hesitation in deciding to join Arsenal from Deportivo Saprissa.
"They showed an interest in me when I was moving to Europe but I have been an Arsenal fan since I was a child and that's why I chose them," said the 21-year-old forward, who has loan spells at Lorient, Real Betis and now Olympiakos.
Campbell may not yet have broken through into the Arsenal first team, but he hopes to be a part of Arsene Wenger's plans next season.
"There are a lot of good players in Costa Rica. Many of them like me think only of being a footballer and it was very exciting when some of the most important clubs in the world started to show interest in me.
"As soon as Arsenal came, I wanted to join them and I believe I am ready to play for Arsenal next season.
"I know I have the capacity to play there. I am only waiting for the opportunity. With hard work I know I can get there."
Speaking about his time with Olympiakos, Campbell added: "This has been a very good experience.
"I am trying to do my best, to do better and continue growing as a footballer. It is why I chose Olympiakos. This is a big team with the opportunity to play in the Champions League, the most important competition in Europe.
"The league here is not as competitive as in Spain but it is a good place for me to grow as a player. We have a strong team, with competition for places. Every day I try to learn something from Javi Saviola. The life here is good, the food and the weather."
Campbell is set to gain further experience on the big stage this summer, with Costa Rica heading to the World Cup after an impressive qualification campaign.
They have been drawn in a tough group with Italy, Uruguay and England, but Campbell is upbeat about their chances.
"It is a difficult group but at the same time it is nice because they are such good teams," he enthused.
"We have the same chance as the others. Anything can happen in football and I trust in my team. We can do it."