The 42-year-old Spanish singer-songwriter and the former tennis star are believed to have become the proud parents of twins, one boy and one girl, who were born over the weekend.
According to TMZ.com, the babies are named Nicholas and Lucy, and Anna gave birth to the tots - who mark the first children for the pair - on Saturday (16.12.17) in Miami.
The news may come as a surprise for many fans of the couple, as the managed to keep the pregnancy completely under wraps for nine months.
It might be a while before the tots' grandparents get to meet the new arrivals though, as 'Hero' hitmaker Enrique admitted earlier this year that despite having dated Anna since 2001, the tennis star has never met his chart-topping dad Julio.
The dark-haired hunk confessed: ''They still haven't met, no. They have to meet at some point.''
Enrique also revealed that despite the length of time they've been together, he and Anna have no immediate plans to get married.
However, Enrique claimed they already live the life of a married couple in many ways.
He added: ''We're just as happy. There comes a point when you've been with someone for a certain amount of time that I believe that you are like, married.
''I guess the only difference is that we haven't walked down the aisle.''
Enrique raised eyebrows when he introduced Anna, 36, as his wife during a concert in Russia in 2011.
But the singer subsequently admitted he simply got caught up in the moment, adding that being married or not has no bearing on how much he loves his long-time partner.
Reflecting on the incident, he previously shared: ''It was just meant to be sweet, in the moment. I honestly didn't mean to confuse people. I thought it'd be easier for the audience to understand than if I said, 'my girl'.
''I've never really thought marriage would make a difference. Maybe it's because I come from divorced parents, but I don't think you love someone more because of a piece of paper. And nowadays it's not taboo to have kids and not be married.
''What makes a difference is that you're a good parent.''
They might sound like they're from the 70s, but they way they roll is very 2020.
What's new in the music world this week?
'U Kin B the Sun' is an album rich in texture and depth and one that quite obviously, and unapologetically, plays to Frazey Ford's strengths.
Listen to their new single 'Small Change'.
Everything you ever needed to know about Viking metal.
3TEETH hit Leeds for the first time with support from British artist PIG.
This will make you want to walk amongst the skyscrapers.