If they are happy that everything is normal, they could wait up to two weeks after the due date to induce the Duchess, but it could happen much sooner.
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 which 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 when he was born two years ago, 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 is "a very safe procedure" which is offered to mothers if they are feeling anxious, fed up or uncomfortable once they have passed their due date.
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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 about inducing any time after the due date if the mother is getting fed up.
"We tend not to leave it more than two weeks after the due date, but most mothers are getting pretty fed up once they go a week past their due date. There are other considerations that come into play - if it's a big baby it's sometimes better to induce before it gets any bigger.
"The mother will normally have a scan on or around the due date to make sure the baby is fine and it's safe to press on a bit longer."
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, which is 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.