Speculation is rife that the Duchess of Cambridge may have gone into labour after a royal convoy was reportedly spotted leaving Kensington Place.

As Kate Middleton, who has been staying at London's Kensington Palace with husband Prince William, is expected to transfer to the Lindo Wing of St Mary's Hospital Paddington when she is ready to give birth, many fans have taken it as a signal that the baby is coming.

News of the convoy sighting come hours after it was reported that the 33-year-old royal is now overdue – it had been suggested that she would give birth on 25 April – and now may have her labour induced.

A source told The Telegraph: "The doctors don't normally wait for more than a week.

"They will probably already have been in touch with Kate and talked about which day would suit her best to go in and be induced if the baby doesn't arrive in the next couple of days."

Meanwhile, St Mary's hospital has reportedly launched an investigation after it emerged that cocaine may have been snorted in the toilets just yards from the Lindo ward.

"We take the safety of our patients and staff extremely seriously," a spokeswoman for Imperial College Healthcare NHS Trust is quoted as saying by the paper.

She added that that the building was constantly monitored by "24/7 security" and that they worked closely with police to ensure "the safety of our colleagues, patients and their families".

She added: "If there are signs that a person is abusing substances on one of our sites we will take immediate action to address this."

Although there has been no confirmation of the baby's sex or due date, that has not stopped Brits across the nation taking a punt on all things royal baby, with bookmaker Paddy Power revealing that bets on the birth had surpassed £250,000 ($384,000).

"Royal baby betting is booming, money is pouring in by the pram load. As of yesterday we nudged past £250,000," said Paddy Power spokesman Rory Scott.

"Depending on how long this drags on for we're expecting to turnover maybe a million, maybe nudge past a million, quadrupling that big figure in the next week or so."

Parking restrictions at the Lindo Wing of St Mary's Hospital in Paddington, where the birth is due to take place, have been extended by Westminster council until 5 May, threatening to take the birth right up to within days of the general election on 7 May.