The boy and girl -- named Nicholas and Lucy -- were born on Saturday here in Miami, reports usmagazine.com.
Iglesias and Kournikova kept the pregnancy a secret. They didn't even announce the arrival of their newborns.
On that same day, Kournikova posted three photographs on Instagram, riding on a boat and wearing an oversized windbreaker jacket from Iglesias' tour.
The couple have been dating since 2001 after meeting on the set of Iglesias' "Escape" music video
They have always remained secretive about their relationship and have not been seen together in public for nearly a year.