Desmond, Celtic’s biggest shareholder, told Sky Sports News that the new appointment “must be the right man”.
He said: “We have talked to a lot of managers and they have all been very good. We are excited about the candidates we have.
“Hopefully in the the next four to five days, or maybe early next week, we will be able to make an announcement - if negotiations are concluded.”
The newly crowned champions had earlier released a statement saying they had no preferred candidate for the position.
It is almost a month since Ronny Deila announced he was leaving Celtic.
Rodgers, the former Liverpool manager, is believed to have been interviewed for the position and Celtic are also understood to be considering a number of other candidates including David Moyes, Roy Keane, Malky Mackay and Neil Lennon.
Rodgers was installed as odds-on favourite on Tuesday but the club say they are determined to “take their time”.
A statement released on Wednesday morning said: “Following some speculation regarding the appointment of a new Celtic manager, the facts are as follows:
“The club has been and will be speaking to a number of candidates and we will take our time to go through this process thoroughly and in the proper manner. There is no ‘preferred’ candidate.
“We will continue this process with the aim of identifying a new manager who we believe will deliver success to the club.
Clearly we will not be able to provide daily updates on this issue. However, our supporters will be the first to know when a new manager has been appointed.”