Reports say hundreds of shops in the souk, one of the best preserved in the Middle East, have been destroyed.
Unesco, which recognizes Aleppo’s Old City as a world heritage site, described the damage as a tragedy.
On the third day of a rebel offensive, battles broke out in the Old City and the Arkub district, reports said.
The fire, believed to have been triggered by shelling and gunfire, began on Friday but was still burning on Saturday, reports said.
“It’s a big loss and a tragedy that the old city has now been affected,” Kishore Rao, director of Unesco’s World Heritage Centre, told the Associated Press.
The market stalls lie beneath the city’s towering 13th Century citadel, where activists say regime troops and snipers have taken up positions.
Activists quoted by Reuters news agency said that the presence of snipers was making it difficult to approach the Souk al-Madina, once a major tourist attraction.
Reports estimate that between 700 and 1,000 shops have been destroyed so far.
“It’s a disaster. The fire is threatening to spread to remaining shops,” one activist, Ahmad al-Halabi, told AP.
He said the Syrian authorities had cut off the water supply, making attempts to control the fire more difficult.
Rebels and civilians were working together to limit the fire with a few fire extinguishers, he added.
The fire took hold with speed, fuelled by the many shops’ wooden doors and the clothes, fabrics and leather goods sold inside.
Heavy clashes erupted at several military sites in the city on Saturday evening, Reuters reports.
Fighting was reported at the Neirab military base as well as Bab Antakya, a stone gateway to the Old City.
The Syrian Observatory for Human Rights, a UK-based activist group, said the focal point for fighting was Salaheddin, a rebel stronghold on the south-west side of the city.
State television reported attacks on what it called “terrorist centres” in 10 different locations on Saturday, saying heavy losses had been inflicted.
Though both sides have reported clashes in different parts of the city, the signs are that the rebels simply lack the firepower and the manpower to score a significant breakthrough.
“No-one is actually making gains here, it is just fighting and more fighting, and terrified people are fleeing,” one activist told Reuters.
Activists estimate more than 27,000 people have died in the violence since the uprising against President Bashar al-Assad began last year.
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