More than 100,000 Syrians fled their country in August, the UN says – the highest monthly number of refugees since the conflict began in March 2011.
“These people sought asylum in neighbouring countries,” said a spokeswoman for the UN refugee agency.
The rising numbers emerged as Red Cross head Peter Maurer discussed Syria’s aid crisis with President Assad.
In Aleppo, supplies are said to be dwindling, with access to parts of the city described as impossible.
Five people were said to have killed in the northern province on Tuesday, a day after at least 25 deaths were reported in air strikes the day before.
Opposition activists reported that many people had been wounded by army shelling in the Darat Izza area of Aleppo province on Tuesday morning.
A Syrian army commander has reportedly vowed to recapture the city in 10 days.
The escalating fighting in Aleppo and earlier in Damascus has intensified the refugee crisis in Syria. In August, the UN High Commissioner for Refugees estimated that 30,000 people had fled to Turkey, Iraq, Lebanon and Jordan in the space of a week.
Turkey has taken in more than 80,000 people and there are reports of thousands more waiting at the Syrian border. Jordan says up to 183,000 Syrians have entered the country since the uprising began.
Armed confrontations
The head of the International Committee for the Red Cross (ICRC) spent less than an hour with President Assad on Tuesday morning. He was also due to meet several other senior Syrian officials and visit some of the Damascus suburbs worst affected by violence.
Syrian state television said afterwards that he supported the organisation’s humanitarian work in Syria as long as it remained “independent and impartial”.
ICRC spokeswoman Cecilia Goin told BBC News that the organisation was planning to “scale up its distribution” of food parcels and other aid across Syria.
“The tense and volatile situation is growing,” she said.
“There are armed confrontations persistently in Aleppo and Homs. It’s impossible to get into Old Aleppo. Unfortunately we are not able to help those people in need in those places where fighting is taking place.”
Last week, the ICRC said it had delivered food parcels to 5,000 people as well as 2,000 mattresses and 1,000 hygiene kits but the agency accepted more civilians were in need of help.
Mr Maurer’s meeting comes days before the new UN-Arab League envoy Lakhdar Brahimi is due to pay his first visit to President Assad in Damascus.
Mr Brahimi, who is to make a statement later to the UN General Assembly, told the BBC on Monday that his task was “nearly impossible”.
Hours before US Secretary of State Hillary Clinton was due for talks in Beijing, a Chinese foreign ministry spokesman stressed the need for a political solution.
“The situation is worsening, but the worse the situation, the more unity is needed,” Hong Lei was reported as telling journalists, according to Reuters news agency.
China and Russia have both vetoed a succession of proposed UN Security Council resolutions on Syria.
The opposition Syrian National Council has continued its call for urgent international military intervention but Beijing has warned that it opposes any such step.
According to the Syrian Observatory for Human Rights, a pro-opposition activist group, more than 5,000 people were killed in the country last month.