Madrid-born Enrique had once caused quite a flutter when he introduced Kournikova as his wife during a concert in Anna's native of Russian Federation in 2011.
The boy and girl - named Nicholas and Lucy - were born at the weekend in Miami, Florida, TMZ revealed. Nowadays, the two live a very quiet life on a private island in Miami Beach. Iglesias also posted a photo from the boat of himself.
The pair met on the set of Enrique's "Escape" music video in 2001 and has been together ever since.
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The celebrity couple is known for being notoriously private.
Iglesias and Kournikova kept the pregnancy a secret. "This is a new era!"
At the last spotting, Anna had sparked marriage speculation when she was seen sporting a large diamond ring and a matching eternity band on her wedding finger. "Besides we are just fine with sex and love", he told Latina magazine. "I don't believe you need a piece of paper to show you love someone", he told E!
Congratulations to Enrique and Anna on their growing family. He quipped: "Just promise me I get one hour of "special alone time" with any boy you decide to bring home in the far FAR future".