A source whose children were born at the Lindo Wing said: "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."
Prince George was reported to have arrived three days late, and anyone who bet money on an early birth this time around has already lost their stake.
Pat O'Brien, a consultant obstetrician and spokesman for the Royal College of Obstetricians and Gynaecologists, said inducing labour was "a very safe procedure" offered to mothers if they are feeling anxious, fed up or uncomfortable once they have passed their due date.
He said: "It's down to the individual consultant as to when it's appropriate to induce labour, but because it's such a safe procedure, particularly with women who have given birth before, they tend to be pretty relaxed.
"Most mothers are getting pretty fed up once they go a week past their due date."
Only three per cent of babies arrive on their due date, with two thirds of first babies arriving late and two thirds of second babies arriving early.
Most women who are induced will only need to be given a hormone gel, applied internally to mimic the hormone prostaglandin, which is released naturally during labour. If that does not work, another hormone called oxytocin can be drip-fed.
Prince George was reportedly born three days late when he arrived at the hospital in July 2013.
The joint favourite dates for the royal baby to be born are 26th and 27th, London time, at 6/1 - with 28th, priced at 7/1 and 29th 8/1, according to MailOnline's official betting partner Coral.
The bookmaker is also offering evens on a morning birth, 3/1 on the baby arriving in the afternoon and 2/1 on it being in the evening - while Alice (5/4) and Charlotte (5/1) are the favourite names.
But some punters might look further ahead for their bets - if they believe those who say labour is influenced by the lunar cycle and the arrival of a full moon. The next full moon is on May 4.
The idea is the moon's gravitational pull affects the amniotic fluid in the same way it affects water in the sea and rivers. When Kate gave birth on July 22, 2013, it was the day of a new full moon.
- Daily Telegraph, Daily Mail