The Blues are thought to have beaten both Liverpool and Blackburn Rovers to the 15-year-old's signature.
Celina, who will move to the Etihad Stadium once he has completed his schooling, has said that the move was 'a dream come true'.
"This is a childhood dream coming true," he told Drammens Tidende. "I believe it will be very good for me."
City, who are currently top of the Premier League table, take on local rivals Manchester United at Old Trafford on Sunday.