The vessel, which was undergoing a £25m restoration, is kept in a dry dock at Greenwich in south-east London.
Police are analysing CCTV images which are thought to show people in the area shortly before the fire started.
A Cutty Sark Trust spokesman said much of the ship had been removed for restoration and the damage could have been worse.
Half the planking and the masts had been taken away as part of the project.
Chris Livett, chairman of Cutty Sark Enterprises which is repairing the clipper, said at the scene: "From where I stand there is not a huge amount of damage to the planking that was left on.
"There are pockets of charred planking and some have gone, but it doesn't look as bad as first envisaged."
In Pictures: Cutty Sark fire
The chief executive of the charitable Cutty Sark Trust, Richard Doughty, said: "What is special about Cutty Sark is the timbers, the iron frames that went to the South China Seas, and to think that that is threatened in any way is unbelievable, it's an unimaginable shock."
Following an inspection of the site on Monday afternoon, Mr Doughty said: "Buckling of the hull remains a big fear but until we do the measurements we are not going to know.
"With my naked eye, as far as I have been able to see, the structure of the ship seems to be intact."
Insp Bruce Middlemiss said detectives were looking into the possibility that the fire had been started deliberately.
Special history
Firefighters were called to the scene at 0445 BST and the flames were put out by 0700 BST.
An area around the 138-year-old tea clipper had to be evacuated during the blaze.
"The cause of the fire is now under investigation by London Fire Brigade and the Metropolitan Police," a London Fire Brigade spokesman said.
A number of witnesses have already come forward and the police are urging anyone else who may have been in the area to contact them.
A silver car was seen leaving the scene but police said there is nothing at this stage to link it to the fire.
CUTTY SARK
Greenwich Council leader councillor Chris Roberts said: "This is a devastating blow for what is a truly iconic symbol of Greenwich across the world.
"The Cutty Sark has a unique and special history, which helps to draw millions of visitors to Greenwich every year."
The Cutty Sark left London on her maiden voyage on 16 February 1870, sailing around The Cape of Good Hope to Shanghai in three-and-a-half months.
She made eight journeys to China as part of the tea trade until steam ships replaced sail on the high seas.
The ship was later used for training naval cadets during World War II, and in 1951 was moored in London for the Festival of Britain.
Shortly afterwards, she was acquired by the Cutty Sark Society.
The ship was undergoing conservation work because sea salt had accelerated the corrosion of her iron framework.
Dr Eric Kentley, curatorial consultant to the Cutty Sark Trust, said of the ship: "It can be saved. It's certainly not completely devastated.
"We will put her back together - but it's going to take much, much longer and a lot more money than we originally thought."
Visit London's chief executive James Bidwell said: "The ship's need for vital conservation has put it in the public eye recently and we can only hope that this terrible fire will redouble all our efforts to preserve this wonderful part of London's heritage."
The Duke of Edinburgh is due to visit the Cutty Sark on Tuesday. Culture Secretary Tessa Jowell inspected the remains on Monday afternoon.
The Cutty Sark Trust is appealing for funds to help repair the fire damage and complete the restoration.
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