The Netherlands Under-19 international attracted interest from the Dutch side following spells at Portsmouth and Blackburn Rovers last season.
The 18-year-old defender told the club's website: "I am very happy to be at PSV. I've been very well received here in Eindhoven and I will give all my best to help PSV win the Eredivisie.
"Also I want to become a better player and develop myself. We have a great manager and a young group full of talent. I am really looking forward to the season starting."
Rekik has made one Premier League appearance for City since joining the club in 2011.