The duchess told the proud father: “He’s gorgeous. It makes me feel a little broody.”
The Cambridges already have a young family of three – Prince George, Princess Charlotte and the most recent addition Prince Louis.
When Mr Barr asked: “Baby number four?” Kate could not stop herself from laughing as she said: “I think William would be a little worried.”
When the duchess was pregnant with Louis she jokingly said her husband was in denial about the new baby.
Ladbroke has odds of 5-2 the couple will announce this year they are expecting baby number four.
Jessica Bridge of Ladbrokes said: “Royal fans are already going into overdrive for Harry and Meghan’s baby, let alone William and Kate announcing their brood is expanding once again.”