The 39-year-old, who gave birth on 22 January, posted a black and white photo of her baby clutching onto her finger.
“My new obsession #soinlovewithhim,” she captioned the shot.
Jones announced she had welcomed her son into the world on 26 January by telling her ‘The One Show’ co-hosts Matt Baker and Angela Scanlon live on air.
“I do have some news - our beautiful, healthy little baby boy was born,” Jones said in the call, that was broadcast across the studio.
She said she had been a “bit behind” finding a name for her son, and it seems fans are getting impatient.
“So?” one person commented on the photo above. “Do you have a name for this special chap?”
Another wrote: “Have you chosen a name yet Alex?!”
Another tried to make a guess of the baby’s name, writing: “Awww many congratulations Alex, he must have a Welsh name meaning ‘special one’.”