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, according to BBC News.
"These people sought asylum in neighboring 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, Sept 4, 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.