He had also responded to a follower who had like the tattooed image, writing: "Thanks :) Never too young for some ink haha."
One Twitter user asked Musk the name of their child, to which he responded, "X Æ A-12 Musk".
It was not immediately clear if the new dad Musk was having fun with his followers with this tweet, which appeared to be partially inspired by one of the songs from Grimes' last album - 4ÆM.
It's Grimes' first child – Musk has five sons with his first wife, Justine.
Grimes announced she was pregnant in January via a social media post that showed her with a foetus Photoshopped into her belly.
During a livestream on Twitch in February, she said that the baby's due date was May 4. Musk tweeted that the baby was due on Monday.
Grimes has said that she will allow her firstborn to decide their own gender.
"I don't want to say the gender of the baby... I feel like their privacy should be protected," she said. "I don't think they can consent to being famous or being in public. I don't want to gender them in case that's not how they feel in their life. I don't know, I just feel like it doesn't need to be known."
The couple's first child together comes almost exactly two years after they made their debut as a couple. In May 2018, Musk and Grimes, whose real name is Claire Boucher, took both the business and music worlds by surprise when they appeared together at the MET Gala in 2018 with Grimes wearing a choker shaped like the Tesla logo.
