Council officials and property owners have been counting the cost of the damage, which is expected to run into many millions of pounds.
Experts have decided that there is a need to demolish some of the buildings as soon as possible.
A decision on when this process will get under way is expected to be taken on Monday afternoon.
It now appears almost certain that every building affected by the fire will come down, and that the facades may also be affected.
Edinburgh City Council leader Donald Anderson said: "Our officers have been in touch with Historic Scotland as virtually the entire site is Category A listed.
"We have been taking advice from them and we will try to make sure that we retain whatever historical character we can.
"But there is going to be extensive demolition."
Lothian and Borders Fire Brigade Divisional Officer David Young estimated that the cost of the blaze would run into many millions of pounds.
"A substantial part of the core of the heart of the Old Town of Edinburgh has been destroyed," he said.
The fire is believed to have started at about 2015 GMT on Saturday above La Belle Angele nightclub, located behind the Gilded Balloon comedy venue.
It was among several properties burnt out in the area, which Historic Scotland said was at the periphery of the city's World Heritage Site.
About 150 people were forced to spend two nights in temporary accommodation as firefighters continued to battle against pockets of fires burning within the complex network of business and residential properties.
Drivers have been warned to stay away from the area, where a number of roads remain closed.
Firefighters said on Monday morning that they had the blaze under control.
Crews were still damping down the fire and planned to keep an eye on the seven-storey Hastie's Close building.
"We have put the big fires out but, as this building is such a big building, fires can break out in different areas and spread," said a spokeswoman.
'Serious blow'
"We are likely to monitor the situation for the next two days in order to make sure that everything is safe."
The cause of the outbreak is not yet known.
Cllr Anderson said Saturday's fire was the biggest in living memory.
"We have a gaping hole in the very heart of the city, so this is a very serious blow for us," he said.
Efforts to stop the fire from spreading to other buildings in the area, including the architecturally significant Adam House, appear to have been successful.