Rovers are looking for a successor to Brian Flynn, who guided the south Yorkshire club to the League One title this season after taking over from Dean Saunders in January following his departure to Wolves.
Flynn was handed the reins on a caretaker basis before being given a contract until the end of the campaign and the former scout will now revert back to that role under the new manager.
Chairman John Ryan expects to have a shortlist drawn up by Monday, with the interviewing process to begin later next week.
He said: "We have already had some high-quality applicants and we will sift them down and have our shortlist by Monday. We will then interview the best candidates and then hopefully, there will be an appointment.
"It is a case of the sooner the better, really. But it depends who it is going to be and if he is in or out of work.
"We have already had some high-quality applicants and I am confident that myself and my two other major shareholders - Terry Bramall and Dick Watson - will continue our good track record of picking good managers."