The 22-year-old joined the Gunners in 2011, but has yet to play a game for the club.
He spent last season on loan at Olympiacos and scored during the Greek outfit's 2-0 victory against Manchester United in the last 16 of the Champions League.
His performances for Costa Rica in the World Cup have led to increased speculation surrounding his future, but he expects to be playing at the Emirates when he returns from Brazil.
"I believe I'll be returning to Arsenal for next season," The Mirror quotes him as saying.
"It's been a beautiful World Cup for Costa Rica. It's gone very well. Qualifying first in a very difficult group is important, but now we have to focus on the last 16.
"We weren't concerned about who we got drawn against in the first round, we were simply pleased to be there, but now we are determined to go further."
Campbell has scored 10 goals in his 34 appearances for Costa Rica.