Three weeks into a major offensive to retake all of Aleppo, government troops have captured about 85 percent of territory rebels controlled in the city's east and were continuing to push forward.
AFP correspondents in the city said rebel areas were facing intense bombardment.
In east Aleppo, terrified residents could be seen fleeing in the streets as heavy bombing shook opposition-held areas.
In the city's west, the explosions from rebel bombing were so strong they shook the windows of buildings. Plumes of smoke could be seen rising from the east.
Cornered in a shrinking enclave in Aleppo's southeast, the rebels have asked for a five-day ceasefire.
Western countries have backed the call, and US Secretary of State John Kerry and Russian Foreign Minister Sergei Lavrov kept up talks on Thursday in an effort to halt the bloodshed.
WATCH: Drone footage of the destruction of Aleppo's Old City
In a wide-ranging interview with Syrian daily Al-Watan published Thursday, Assad predicted victory for his forces in Aleppo, though he admitted that would not end the country's conflict entirely.
"It's true that Aleppo will be a win for us," Assad said.
"Let's be realistic -- it won't mean the end of the war in Syria," he said. "But it will be a huge step towards this end."
Rebels seized control of large parts of Aleppo in 2012, dividing Syria's former commercial hub into an opposition-held east and government-controlled west.
Red Cross evacuates civilians
The battle in Aleppo Source: AFP
For years Aleppo was a key battleground and important rebel stronghold, but Assad's forces have recently made a concerted push to retake the city.
In the last week government forces steadily gained ground until on Wednesday -- after a highly symbolic retreat from the Old City -- the rebels called for the ceasefire to allow thousands of civilians to evacuate.
Assad's government has said a truce is only possible after a full rebel withdrawal, and opposition fighters have rejected any talk of abandoning Aleppo.
On Thursday the army, backed by fighters from Iran and Lebanon's Hezbollah movement, was continuing to advance, said the Syrian Observatory for Human Rights.
The Britain-based monitor said there were heavy clashes in several rebel neighbourhoods, including Bustan al-Qasr, Saif al-Dawla, Zibdiya, Sukkari and Kallaseh.
All rebel areas were under heavy bombardment, it said, and opposition forces were returning fire with rockets into west Aleppo.
WATCH: Syrian army captures part of rebel-held Aleppo
At least 384 civilians have been killed in east Aleppo during the offensive, while rebel fire into the west has killed at least 105 people, the Observatory says.
The assault has prompted a mass exodus from east Aleppo where at least 80,000 people have fled their homes, according to the monitor.
On Thursday, hundreds of families, most of them from the Salhine district, arrived in Aziza, a southeastern suburb of Aleppo, after the latest rebel defeats.
"I feel reborn," said Yasser, a 40-year-old father of four as he tugged at a cart stacked with luggage atop of which sat his ailing mother.
The UN renewed its call for an immediate ceasefire in Aleppo, warning as many as 500 sick and injured children desperately needed to be evacuated.
"There has to be a pause," said Jan Egeland, head of the UN-backed humanitarian taskforce for Syria.
"At the moment, those who... try to escape are caught in crossfire, they are caught in shelling, (and) risk being hit by snipers."
WATCH: SBS reporter Luke Waters sat down with Assad in July
The International Committee of the Red Cross said it carried out an operation on Wednesday night with Syria's Red Crescent to evacuate 150 civilians, many disabled or sick, from a health facility in the Old City.
It was unclear how many civilians remained in rebel territory, but there were an estimated 250,000 in east Aleppo prior to the latest offensive.
An appeal also came from the White Helmets rescue group for international organisations to protect its members in rebel-held parts of east Aleppo.
"If we are not evacuated, our volunteers face torture and execution in the regime's detention centres," they said in a statement.
Diplomatic efforts
A series of diplomatic efforts to stop the fighting this week have stalled, with Moscow and Washington trading accusations of blame.
Moscow is a key Assad ally and launched an air war in support of his forces last year, while Washington and other Western nations have supported rebel forces.
Russia this week suggested a deal was in the works for rebels to be allowed to withdraw from Aleppo to other opposition-held territory.
Kerry and Lavrov met twice Thursday in the German city of Hamburg but without any announcement of progress on a ceasefire.
Their talks were held on the margins of a meeting of the 57-member Organisation for Security and Cooperation in Europe.
