The 21-year-old has impressed on loan at Olympiakos this season and enhanced his burgeoning reputation with a magnificent goal against the Premier League champion as the Greek side secured a 2-0 win in the first leg of their Champions League last 16 tie.
Speaking ahead of Tuesday’s match, Campbell told Sky Sports: "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. United 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."
Campbell has previously spent loan spells at Lorient and Real Betis following his move to Arsenal from Saprissa and has scored eight times for Costa Rica in 30 senior international matches.