According to Betfair, Alice, Celia and Augusta top the list of potential girls names, while Arthur, Henry and Godfrey are popular choices for a boy.
Betfair spokesperson Darren Hughes said: "Princess Eugenie… announced her first pregnancy, and as always with a royal baby announcement, speculation has turned to what she might name her child.
"Punters seem to think that a traditional royal name is on the cards, and Alice and Arthur lead the betting as 8/1 joint-favourites.
"Other names in the running for a girl include Celia at 10/1 and Augusta at 14/1, with Diana, after her late aunt, at odds of 20/1. Popular boys names in the betting include Henry at 10/1 and Thomas at 14/1, while Boris is an unpopular-looking 200/1. It appears unlikely, however, that the princess will name the child after her sibling, as Beatrice, which is currently at odds of 100/1."
Want to see the full list of possible names for the couple’s first child? Take a look…
Princess Eugenie Child gender odds
Princess Eugenie baby name for a girl
Princess Eugenie baby name for a boy
Will the royal couple choose a traditional name or go for something a little different? Only time will tell!
This article originally appeared on marie claire UK.