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.