"Jerry is a strong name, absolutely", William said.
Since giving birth on Monday, 23 April, Arthur has remained the firm favourite (at 2/1), closely followed by James (4/1), Albert (6/1), Philip (8/1) and Alexander (10/1), reports Time.
Royal watchers were quick to point out the Prince George's middle name is Alexander, suggesting that it would be off-limits as it's already been used.
The gender of the baby hasn't been announced.
The names of royal babies are historically not announced straight away, leaving the nation playing a guessing game until the chosen one is finally revealed.
Prince William also appeared to drop a huge hint that the couple may have gone for another name.
Whatever his name, the new baby's official title will be styled the same as his siblings: Prince (name) Cambridge. Just seven hours following his birth, the proud parents posed for photos on the steps of the hospital cradling their newborn.
When asked if they had considered the name Alexander, he said: "Funny you should say that..."
"It is tradition that a framed notice of birth goes on display on a ceremonial easel on the forecourt at the Palace", the statement continued.
And look at Princess Charlotte's little wave. The royal wedding of Prince Harry and Megan Markle is taking place on May 19.
The duke and duchess of Cambridge have spent their first day at home as a family of five.