Bookmakers have seen George and James become the favourites for the new prince's name.
George was the favourite with William Hill (5/2) and Paddy Power (6/4), while Ladbrokes had James at 2/1.
Henry, the given name of the Duke's brother Prince Harry, is also in the running, along with Alexander and Louis.
Ladbrokes has Henry at 5/1, the same as it is offering George, after what it said was an "unprecedented amount of bets" in the last 48 hours bringing it in from 50/1.
The public may have to wait to cash in their betting slips. It is not uncommon for royals to take their time naming babies: Queen Elizabeth II and her husband Prince Philip waited for a month in the case of Prince Charles.
The latest addition to the royal family has been named Prince George Alexander Louis, but where did the Prince get his names?
The Duke and Duchess of Cambridge have named their son George Alexander Louis, Kensington Palace said.