Arthur, Albert and Philip are all thought to be in the running for the new royal baby's name.
As is tradition, the Duke and Duchess of Cambridge have not immediately announced the chosen name of their third child and second son.
They are likely to wait until they have informed the Queen and the rest of their family.
Historian Judith Rowbotham predicted William and Kate would pick Philip for the new baby prince, in tribute to the Duke of Edinburgh.
"I strongly suspect that Philip will be either a first or a second name," she said.
"It's plain that they like the Germanic sounding names but at the same time they're looking within the Windsor traditions."
Arthur has been the bookmakers' favourite for a boy, followed by Albert.
Catherine, Duchess of Cambridge departs the Lindo Wing of St Mary's Hospital with her newborn baby son on April 23, 2018 Source: Getty Images Europe
Here are some of the contenders:
Arthur
One of Charles's middle names, Arthur is also one of William's middle names and was a middle name of the Queen's father, George VI.
The legendary King Arthur was the mythical leader of the knights of the Round Table, who supposedly lived in the 5th or 6th century.
Once popular, the name fell out of fashion but has had a revival in recent years. Former prime minister David Cameron has a son called Arthur.
Albert
Queen Victoria used to insist that the name Albert was used as a middle name by her descendants, if not a first, in honour of her much-loved consort Prince Albert.
By choosing Albert or Bertie for a boy, William and Kate would be honouring Queen Elizabeth II's father, George VI, who was actually Albert Frederick Arthur George but always known to his family as Bertie.
Shy, stammering Bertie was forced to become king when his brother Edward VIII abdicated, but won the nation's affection by standing firm in London during the Second World War.
Philip
A lasting tribute to the Duke of Edinburgh might see a Prince of Cambridge called Philip.
Both Charles and William have Philip as a middle name.
The Duke - known for his dedication to duty and his acerbic wit - has been married to the Queen for more than 70 years and is the nation's longest serving consort.
Frederick
A Prince Freddie of Cambridge would have a historical link to the 1st Duke of Cambridge.
Prince Adolphus Frederick lived from 1774 to 1850 and was a son of George III.
He was apparently very fond of interrupting church services by bellowing out "By all means" if the priest said "Let us pray".
Charles
William may want to pay tribute to his father - but perhaps as a middle name as it may be considered too similar to Charlotte.
