He is understood to be one of a short-list of four which will be discussed by Smith's selection committee either later this week or next.
The other three are all internal candidates - Alan Giles, managing director of Waterstone's; Richard Handover, who runs the WH Smith News distribution arm; and John Hancock, chief executive of the American division.
It emerged over the weekend that Keith Hamill, finance director, has pulled out of the running. Within the last week he has been at the centre of controversy surrounding an alleged bid to garner support for a break- up of the stationery to record and book shops group.
It is also clear now that Mr Cockburn, whose shock decision to quit the ailing retailer was announced in June, has already left the business and will not be returning.
Mr Cockburn was supposed to remain at the company until October when he joins BT as head of its domestic operations. But he left at the beginning of August for a month's holiday and will not return.
Insiders says the absence of a chief executive, even one who was about to leave, has left the company without strategic direction.
Jeremy Hardie, WH Smith's chairman, has been running the company, though he is only part-time and the vacuum is thought to have left the three internal candidates for Mr Cockburn's post jockeying for position as factions form behind each of them.
The company says Mr Hardie has not been distracted by the search for a new chief executive as this has been led by the nominations committee.
That consists of three of the group's non -executive directors: Martin Taylor, chief executive of Barclays Bank, Marjorie Scardino, chief executive of Pearson, and Patrick Lupo, an American who is chairman of DHL Worldwide.
The company will not make an announcement about Mr Cockburn's replacement at its full-year results meeting next Wednesday.
However, it is thought that the successful candidate will be named next month. "We aim not to disappoint," the company said.