The Bhoys' majority shareholder is hoping to name the club's new boss in the near future.
The Irish billionaire refused to speculate on the identity of Ronny Deila's successor, but admitted that Brendan Rodgers is a "great manager" and "worthy candidate" for the role.
Rodgers, the ex-Liverpool manager, was reported to have held talks with Celtic this week, along with former Bhoys defender Malky Mackay.
Three more with Parkhead connections, David Moyes, Roy Keane and Neil Lennon, all feature highly on the bookies' list of favourites, and Desmond confirmed the Scottish champions have spoken to "a lot" of candidates.
"Hopefully in the next four or five days, maybe early next week we will be able to make an announcement," Desmond said, as quoted by the Daily Mail. "Negotiations are currently ongoing. We have spoken to a lot of managers.
"There’s lots of people. I wouldn’t speculate on who the person is just now. All the people we have interviewed merit the job. (The club has interviewed) in excess of six."
Earlier this week, the Daily Record reported that Rodgers is being paid £5 million per season as part of a severance package agreed with Liverpool upon his sacking in October 2015.
However, it is claimed those payments will cease as soon as the Northern Irishman signs a contract elsewhere, making him a potentially costly appointment.
Rodgers grew up a Celtic fan and credits the late former Bhoys midfielder and manager Tommy Burns with handing him his first break in coaching.