The United Nations says more than 100,000 Syrians fled the country in August - the highest monthly number of refugees since the conflict began in March last year.
A spokeswoman for the UN's refugee agency the people sought asylum in neighbouring countries.
The rising numbers emerged as Red Cross head Peter Maurer discussed Syria's aid crisis with President Bashar al-Assad, the BBC reports.
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.
Mr Maurer's meeting comes days before the new UN-Arab League envoy Lakhdar Brahimi is due to pay his first visit to Mr Assad in the capital 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".
