Syrian rebels have retreated from former strongholds in eastern Aleppo in a "terrifying" collapse, holding onto a small sliver of territory packed with fighters and thousands of civilians as government troops press on with their rapid advance.

The Syrian military said it had gained control of 99 per cent of the former opposition enclave in eastern Aleppo, signaling an impending end to the rebels' four-year hold over parts of the city as the final hours of battle played out.

"The situation is very, very critical," said Ibrahim al-Haj of the Syrian Civil Defense, volunteer first responders who operate in rebel-held areas.

Retaking Aleppo, which has been divided between rebel- and government-controlled zones since 2012, would be President Bashar Assad's biggest victory yet in the country's civil war.

But it does not end the conflict: significant parts of Syria are still outside government control and huge swaths of the country are a devastated wasteland. More than a quarter of a million people have been killed.

On Sunday, the Islamic State group re-occupied the ancient town of Palmyra, taking advantage of the Syrian army and its Russian backers' preoccupation with the fighting in Aleppo.

In Aleppo, staff members of the last remaining clinic in rebel-held territory huddled in a shelter as Syrian government forces pushed in. "Those killed and wounded are left on the streets," said the clinic's administrator, Mohammed Abu Rajab.

"The collapse is terrifying," said Bassam Haj Mustafa, a rebel spokesman in contact with fighters in the city. Opposition fighters were "doing their best to defend what is left," he said.

Rami Abdurrahman, who heads the Britain-based Syrian Observatory for Human Rights said more than 60 civilians and fighters were killed in rebel-held neighbourhoods of Aleppo on Monday alone.

Russia, a key ally of Assad, refused an American proposal for a temporary halt to the fighting to allow the safe departure of those remaining in rebel-held areas, said a US official.