According to the French La Chaine Info television, looting and fires followed intensive fighting between supporters of Foundi Abdallah Ibrahim, the self-proclaimed president and his former prime minister, Chamasse Said Omar.
Mr Foundi, who is seeking full independence from the Comoros, unilaterally declared independence for Anjouan last year, while Mr Chamassi is calling for a return to rule by France, the former colonial power.
France says it has had no request to intervene while the Organisation for African Unity is considering sending in troops to halt the fighting.
Fighting began on Saturday and continued even after a reported Sunday ceasefire
Latest reports say up to 60 people have died in machine gun and artillery battles while thousands have fled their homes.
According to one report, Mr Foundi's supporters in Mutsamudu used rocket launchers to attack the neighbouring village of Mirontsy.
Residents in Anjouan said fighting died down on Tuesday, but La Chaine quoted witnesses who said that "summary executions of civilians" were taking place.
The Government of the Comoros islands earlier claimed that France, the former colonial power, was ready to provide logistical help.
An exiled Comoros opposition leader has also appealed to France to help bring about negotiations to end the fighting.
"Our people are desperate," said Achirafi Said Hachim, head of the Rally for Democratic Renewal in Comoros.
"I want to launch an urgent appeal to France to help our people find unity and rebuild their future."
Mr Hachim, an exile in France, called for for Paris to back the Organisation of African Unity (OAU) to restore peace on the island.
Mamadou Kane, of the OAU, said: "The situation is catastrophic. Thousands of people have quit their houses and their villages."
OAU ambassadors in Addis Ababa called on the region "to assist in the restoration of peace, security and stability".
One-time allies
Mr Foundi and Mr Chamassi both originally advocated a return to French rule, along the lines of the nearby island of Mayotte.
Mayotte voted against joining the other islands of the archipelago when they became independent in 1975 and is still administered from Paris.
But Mr Foundi has since decided to press for full independence, saying that a return to French administration was not feasible.
Both leaders have said they will boycott presidential elections which are due following the death last month of President Abdulkarim Mohamed Taki.
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Comoros president dies
(06 Nov 98 | Africa)
Comoros information
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