William Hill is offering odds of 7/4 that the Duchess of Cambridge will give birth this weekend, while an analysis of social sites suggests most people think the offspring will be a girl.
"There is more media outside the hospital than ever, rumours keep circulating on social media – levels of expectation are at fever pitch,” William Hill spokesman Joe Crilly said.
On Thursday Twitter was awash with rumours that Kate Middleton had gone into labour, although the rumour was quickly denied by the palace.
The bookmakers also saw a brave punter stake £1,000 on the new arrival being a boy.
Meanwhile Synthesio, a global social media monitoring company, said 62% of social media users predict that the royal baby will be a girl.
The most popular name is Alexandra, followed by Elizabeth, Margaret, Victoria and Charlotte.
For a boy, users predict that he will be called George, with James, Andrew, Philip and Edward also in the running.
Synthesio looked at conversations on Twitter, Facebook, blogs, forums and comment sections of online publications around the world.
Across the US and the Commonwealth countries, the most popular girl’s name is Elizabeth, whereas George is the favoured boy’s name across the globe.