Edinburgh, 46, was dismissed on Tuesday with the Gills 17th in the League One table and Scally unhappy with their form during 2016.
"I am not even convinced I am going to go down the manager route as we are looking at other alternatives now," Scally told BBC Radio Kent.
"We may have a head coach and a team of coaches, with a director of football."
LISTEN: Gillingham chairman Scally on Edinburgh sacking
He added: "There are lots of very good coaches out there who can do specific parts of the work.
"There are lots who have done well at different levels and we are looking at a combination at the moment."
Gillingham had been in the automatic promotion places in February last year, but a slump in form led to the Kent club finishing last season ninth in the table, five points off the play-offs.
Their inconsistent form continued this campaign, with Scally criticising performances following the FA Cup defeat by non-league Brackley Town in November.
The Priestfield Stadium side won just 13 of their 46 league games in 2016, and are eight points above the relegation zone.
"The team is the most expensive one we have ever had under my reign, and the club's history," Scally said.
"It isn't performing to the level I think it should be. If we carry on with the current performance there is a chance we will get dragged into a relegation battle."
Scally, who returned to work this week after two months recovering from a heart operation, says he will have the managerial situation addressed "very quickly".
Former Gills defender Ady Pennock is among the bookmakers' favourites to fill the vacancy with the first team next in action at Oldham on Saturday, 14 January.
"I am feeling fairly good and strong but it is not the start to the New Year and my return that I wanted," Scally added.
"There will be lots of names put forward, but people who have had a taste of Premier League and Championship football want to stay there.
"There are lots of options and we have a number of applications, as always. There is no cheap option."
Gillingham chairman Paul Scally may change the club's management structure after sacking boss Justin Edinburgh.