“Sweetest Christmas gift,” said one.
“So happy for the two of you, the family just keeps growing,” wrote another.
“May this little soul bring endless joy, light, and purpose to your days, and may your family be wrapped in health, peace, and unforgettable moments,” added a third.
Anna Kournikova is a former tennis star, model and TV personality.
Kournikova and Iglesias have been together since 2001 and are already proud parents to twins Lucy and Nicholas, aged 7, and Mary, 5.
News that the couple were expecting again first broke in People in August.
“Anna and Enrique are very happy to become parents again,” People reported.
“They have proven to be great parents to their three children, and both of them love the process.
“They enjoy the activities and all that goes along with raising children.”
Anna Kournikova and Spanish singer Enrique Iglesias met on the set of his Escape music video. Photo / Justin Campbell/BuzzFoto/FilmMagic
Concerns for Kournikova’s wellbeing were raised earlier this year when she was spotted wearing a large leg brace and using a wheelchair.
The Daily Mail reported she may have undergone surgery for a foot injury at that time.
The Russian tennis star and model met Iglesias on the set of his Escape music video.
People reported the singer has modified his demanding touring schedule to allow him more time with his family.
“Being a dad is very important to him,” their source said of Iglesias.
Kournikova achieved a world No. 8 ranking in 2000, and was a Wimbledon semi-finalist in 1997, losing to eventual winner Martina Hingis.
A series of spinal injuries set her career back and she retired in 2003.