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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."
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 Built in 1869 at Dumbarton on the River Clyde Designed by Hercules Linton First voyage February 1870 210ft (64m) long Main mast stood 152ft (46.3m) above the deck Has had 15 million visitors Preserved as a tribute to merchant navy workers
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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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The Cutty Sark in January 2005.The Cutty Sark, one of the most famous historic sailing ships in the world, was seriously damaged by fire in the early hours of Monday morning, May 21, 2007. The 19th century ship, which is in dry dock in Greenwich, London, England, set a speed record during its working days, and has been a popular tourist attraction for many years.

The fire brigade was first called to the blaze on the tea clipper at 4:45 a.m. BST and reported that the flames had been extinguished by 7:00 a.m., but firefighters were still on the scene damping down at 8.30 a.m. The blaze was so intense that eight fire engines and 40 firefighters were sent to put out the fire. No injuries have been reported. 

Cutty Sark Trust chief executive Richard Doughty said he was told the blaze was being treated as suspicious. "We're losing history," he lamented. "It's unbelievable." He went on: "When you lose the original fabric, you lose the touch of the craftsmen; you lose history itself. What is special about Cutty Sark is the timber, the iron frames, that went to the South China Sea. To think that is threatened in any way is unbelievable. It is an unimaginable shock." 
The Cutty Sark being restored, one week before the fire. 
However, Doughty also confirmed that half of the ship's planking, and all of the masts and rigging had been removed for renovations prior to the fire. The iron framework of the hull has been twisted and buckled.

The Chairman of Cutty Sark Enterprises, Chris Levitt, speaking later at the scene, said: "We had removed 50% of the planking, so 50% of the planking wasn't on site and that's safe and secure, and 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."

Cutty Sark Trust curatorial consultant, Dr Eric Kentley was optimistic that the Ship is not completely devastated and can be saved. Speaking about the Cutty Sark, he said: "We will put her back together - but it's going to take much much longer and a lot more money than we originally thought." 

The historic ship, one of London's best known tourist attractions, has been in dry-dock in Greenwich since 1954. It is currently undergoing a £25 million renovation scheduled to last until 2009. It was feared that gas canisters used in the reconstruction work might explode in the blaze but the London Fire Brigade later confirmed that none were present, although concerns about the risk of explosion had caused some delay in tackling the fire. It is a Grade I listed monument and is on the ''Buildings At Risk Register'' in the UK.
The ''Cutty Sark'' under full sail with the reduced rigging, taken by her captain R. Woodget from two joint boats between 1885-1895
===Arson being considered===
Police believe the fire on board the ship may have begun in suspicious circumstances. They are studying CCTV footage which showed some movement around the ship in the early hours of the morning. They are appealing for witnesses who might have seen anyone near the ship or a silver car that was reported driving away from the scene.

=== Last survivor ===
The Cutty Sark was built in 1869 by Scott & Linton, Dumbarton in Scotland and completed by Dennys. It is the world's last surviving tea clipper. Because the clippers were build of wood, but with a framework made of iron, they marked the transition from wood to iron. The windjammers that emerged some 30 years later were build completely of iron or steel.

The clippers were used in long-distance races between China and England, with large profits for the first ship back with the first tea of the year. The Cutty Sark sailed in the China trade between 1870 and 1877/78. She sailed in the Australian wool trade between 1883 and 1895, during which period she achieved the record breaking voyage under wind power between Australia and England via Cape Horn of 72 days in 1885.

The ship takes its name from a character in Burns' poem 'Tam o' Shanter''.