Ladbrokes has given Barnes odds of 4/1, putting him top of its all-male top five candidates, with just one woman, Harriet Green, in the top 10.
Barnes is currently joint group managing director for Aldi UK and Ireland with Ronan Heini, but is slated to take over as chief executive officer in April.
Tesco's former Asia chief, David Potts, is another favourite at 5/1, sharing second place with the man who turned around Monsoon Accessorize, John Browett.
B&Q's commercial director, Darren Blackhurst has longer odds of 8/1. Mark Harrison, Morrisons' current manufacturing director, has dropped from being a favourite to odds of 10/1.
Former Thomas Cook chief executive officer Harriet Green is 12/1. The only other woman on the list is Eleanor Morrison, daughter of former chairman Sir Ken Morrison, at 200/1.
Morrisons' ad stars Ant and Dec have a 1000/1 shot at the top role.