James and Philip have become the new favourite names at the bookmakers for Princess Eugenie’s baby boy.

Eugenie and Jack Brooksbank’s son – the Queen and the Duke of Edinburgh’s ninth great-grandchild – was born on Tuesday at the exclusive Portland Hospital, weighing 8lb 1oz.

The couple, who are settling into life as a new family of three, have yet to reveal what their baby will be called.

Bookmaker Coral said James has overtaken Arthur in the name stakes, while Betfair has Philip – after the baby’s 99-year-old great-grandfather – as the new favourite.

The royal family traditionally wait until the Queen has been informed of a baby’s name before making it public.

The monarch and the Duke of Edinburgh are said to be delighted at the new arrival, as are first-time grandparents the Duke of York and Sarah, Duchess of York.

Odds from Coral that the Queen’s granddaughter and Mr Brooksbank have chosen James are 2-1, while Arthur is 3-1, with Oliver at 6-1 and Edward 8-1.

Coral’s John Hill said: “James has leapfrogged Arthur in our baby name betting on Princess Eugenie’s boy.

“It’s obviously a name which carries a lot of royal tradition so it is no surprise that it is the favourite ahead of the likes of Arthur and Oliver.”

James is a Stuart name. James I, son of Mary, Queen of Scots, had been king of Scotland for 36 years as James VI when he became king of England in 1603.

Eugenie already has a cousin James – the Earl and Countess of Wessex’s 13-year-old son, Viscount Severn.

But Betfair has Philip at 4-1 with Arthur at 5-1, and the more unusual Godfrey also at 5-1.

Betfair spokesman Sam Rosbottom said: “Royal punters are backing them to name their new bouncing baby boy after his great-grandfather.

“Philip was 25-1 but is now 4-1 favourite, having been slashed overnight.”

Ladbrokes still has Arthur as favourite, reducing the odds from 5-1 to 4-1.

Arthur evokes the legend of King Arthur – the mythical leader of the knights of the Round Table.

It has been a popular middle name for royal boys during the Windsor era –  but not a common first name.

Ladbrokes also reported a “flurry of weird and wonderful” bets.

Spokesman Alex Apati said: “The betting suggests Eugenie and Jack will choose a traditional royal name, although a handful of punters seemingly reckon the happy couple will opt for something a little more out-there.”

Tarquin is 12-1, while Humphrey is 16-1 and Horatio and Quentin are 25-1, Barnaby and Axel are set at 33-1, with Boris an outside gamble at 200-1.

When Eugenie was born in 1990, the Duke and Duchess of York’s unusual choice of name for their second daughter took everyone by surprise.

Pronounced “Yoo-junnee”, it was inspired by one of Queen Victoria’s granddaughters, Victoria Eugenie of Battenberg, and the princess goes by the nickname “Euge”.

Eugenie, 30, and Mr Brooksbank gave the world the first glimpse of their baby on Tuesday, posting a black and white image on Instagram of their hands cradling his tiny fingers and wrist.

Buckingham Palace said both the princess and the baby are doing well, and that Mr Brooksbank was at his wife’s side for the birth.

The Duchess of Sussex also had her son Archie at the plush, private Portland Hospital in central London, and Eugenie too was born there, as was her sister, Princess Beatrice.

The baby is 11th in line to the throne, and his arrival means that the Queen and Philip’s youngest son, Edward, Earl of Wessex, who was third in line when he was born in 1964, has moved down to 12th place in the line of succession.