The Wales international broke the news on his Twitter account, sharing a photograph of his newborn son.
We are delighted to welcome a baby boy into our family this morning. Axel Charles Bale 08.05.18 👶🏻💙 pic.twitter.com/dFbHNdFwnH— Gareth Bale (@GarethBale11) May 8, 2018
"We are delighted to welcome a baby boy into our family this morning. Axel Charles Bale 08.05.18," the winger tweeted.
Bale and Rhys-Jones, who are due to be married this month, also have two daughters together - five-year-old Alba Violet and Nava Valentina, who is two.
The former Tottenham Hotspur midfielder's future at Real has been cast into doubt following a tough season in Spain, with Manchester United and Inter Milan touted as potential summer destinations should he choose to move on.