Syria's military and Russian warplanes bombarded rebel-held districts of Aleppo on Saturday as Damascus's allies said victory was near, but insurgents fought back and army advances halted after rapid gains during the week.

The United States said it was meeting a Russian team in Geneva to find a way to save lives, but an agreement looked elusive as the two countries, which back opposing sides, have repeatedly failed to strike a deal to allow evacuations and help aid deliveries.

Russia, whose military intervention helped turn the war in President Bashar al-Assad's favour, said the Syrian government now controls 93 per cent of Aleppo, a figure Reuters could not independently verify.

Recapture of the country's second-largest city would deal a major blow to rebels who have fought to unseat Assad in the nearly six-year war. The insurgents are holed out in a handful of areas mostly south of the historic Old City, having lost nearly three-quarters of territory they controlled for years in the space of around two weeks.

'Promised victory'

Lebanese Shia group Hezbollah, a key military ally of Damascus alongside Russia and Iran, said late on Friday that a "promised victory" in Aleppo was imminent and would change the course of the war.

The advances mean the government appears closer to victory than at any point since 2011 protests against Assad evolved into armed rebellion.

The war has killed more than 300,000 people and made more than 11 million homeless. A win for Assad in Aleppo looks close, but the fighting still raged on Saturday.

Government loses most of Palmyra

In a surprise setback elsewhere, government forces lost control of most of the ancient city of Palmyra in eastern Syria to the Islamic State in Iraq and Syria group, a war monitor and rebels said.

The army earlier said it had sent reinforcements to Palmyra, more than 200 kilometres away, to stave off a fierce attack by the militants. A rebel commander in the Aleppo-based Jaish al-Mujahideen group said the ISIS offensive had forced the government to divert troops to Palmyra from Aleppo — a possible explanation for the slowed advance there and heavy aerial and artillery bombardment.

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Russian warplanes and Syrian artillery bombarded rebel-held districts, and rebels responded with shelling of government-controlled areas as gunfire rang out, a Reuters correspondent in Aleppo said.

Russia and Syria said on Friday they had reduced military operations to allow civilians to leave. But rebels said their counterattacks are what have halted government advances.

"There's no advance by the regime. They (rebels) have stopped them several times," Zakaria Malahifji, a Turkey-based official in the Fastaqim rebel group said.

Vast destruction

Fighting has killed hundreds of people in recent weeks, monitors say, and devastated large areas of east Aleppo.

Parts of the UNESCO World Heritage Old City recaptured by the government were completely destroyed by fighting, a Reuters correspondent said.

I found my home destroyed. I didn't even recognize where it was because of the destruction. - Aleppo resident

Old markets and bathhouses had been flattened.

"I found my home destroyed," said one returning resident, who gave only his family name, Sheikho.

"I didn't even recognize where it was because of the destruction," he said.

'They destroyed civilization'

Mohammed Shaaban, standing outside a destroyed church, was also astounded by the destruction.

"A year and a half ago when I last visited there was not this level of damage. I'm shocked and saddened. They destroyed civilization and humanity," he said, referring to rebels.

Boys collect firewood to be used for heating in a rebel-held, besieged area of Aleppo on Saturday. (Abdlrhman Ismail/Reuters)

The Syrian Observatory for Human Rights monitoring group said several people were killed in rebel shelling on Saturday. Thousands of people have left rebel districts. Some have fled to government-held areas but others went to areas under rebel control fearing arrest and reprisals by government forces.

Kerry urges Russia to show 'a little grace'

U.S. Secretary of State John Kerry urged Russia to show "a little grace" when American and Russian officials meet in Geneva later on Saturday to try to reach a deal enabling civilians and fighters to leave Aleppo.

U.S. Secretary of State John Kerry met leading diplomatis in Paris in an effort to find solutions for the conflict in Syria. (Thibault Camus/Associated Press)

"Fighters ... don't trust that if they agreed to leave to try to save Aleppo that it will save Aleppo and they will be unharmed," Kerry told reporters in Paris after a meeting of countries opposed to Assad.

"The choice for many of them ... is to die in Aleppo, die in [neighbouring] Idlib, but die," he added.

Germany said Syrian opposition backers were seeking a political solution, but there was no agreement in Paris on a truce.

Russia's defence ministry said more than 20,000 civilians left eastern Aleppo on Saturday and over 1,200 rebels laid down their arms. The British-based Observatory said hundreds of civilians had left but no fighters surrendered. Rebel officials have sworn they will never leave.