The front-runners include more traditional names, such as Diana, Grace and Alice but bookmakers think Harry and Meghan could opt for an Italian moniker, if their newborn is a girl.
According to Ladbrokes, Allegra is now the sixth most likely name in the royal baby betting after a flood of bets over the last 24 hours have seen the bookies slash odds from 100/1 down to just 12/1.
Alex Apati of Ladbrokes told Mail Online: 'We're scratching our heads as to why we've seen so much interest in Allegra, but the bets are coming in thick and fast and it's been by far the most popular pick of the month with punters.'
Allegra, which means joyful and lively, is reportedly one of the names Harry’s mother Princess Diana loved.
A report by the Evening Standard in 2004 claimed Diana wanted a baby daughter with her surgeon lover Hasnat Khan.
The princess told a confidante that she wanted to call her baby Allegra after her friend Lady Annabel Goldsmith mentioned the name, because she thought it was “beautiful.”
Diana and Grace are currently 6/1 on royal baby name odds, while Arthur and Elizabeth are at 8/1.
Other favourites if the child is a boy also include James, Edward and Philip.
Some fans are convinced they have discovered the infant’s name and gender thanks to a telltale glitch on the Royal Family website.