It has been reported that Jenna Coleman, who plays Clara Oswald, will leave the hit BBC show at Christmas and, according to Ladbrokes, Leslie, 27, is favourite to replace her, with odds of 14/1.
The BBC has refused to comment on rumours of Coleman's departure, saying people will have to watch the new series to see if she will return or not.
Leslie would join fellow Scots Peter Capaldi and show runner Steven Moffat on the show if she won the role.
Other Game of Thrones stars touted for the part include Alfie Allen, Jack Gleeson and Maisie Williams with odds of 20/1.
Odds of 16/1 are also being offered on Karen Gillan returning to the show, while the same odds are given for Chris Addison taking on the role.
Scot John Barrowman is being given odds of 20/1 to take on the part and return as captain Jack Harkness.
The new series, the first starring Capaldi as the Doctor, begins on BBC One on Saturday.