Lebanese group at risk of losing one of its two closest allies as Syrian President Assad struggles to crush uprising.
The UN refugee agency says 100,000 refugees fled Syria in August alone, the highest monthly total since the uprising began.
Melissa Fleming, spokeswoman for UNHCR, said on Tuesday in Geneva, Switzerland, the rise in people seeking asylum in neighbouring countries brings the total of Syrian refugees registered or awaiting registration with her agency to 234,368 as of September 2.
“When you do the math, it’s quite an astonishing number,” she said.
The rising numbers emerged as Peter Maurer, head at the International Committee of the Red Cross (ICRC), discussed Syria’s aid crisis with President Basher al-Assad.
In Aleppo, supplies are said to be dwindling, with access to parts of the city described as impossible.
Activists say about 5,000 people were killed in August, the highest toll in the 17-month-old uprising and more than three times the monthly average.
Turkish authorities say they have taken in 80,000 refugees, while Jordan says up to 183,000 Syrians have entered their country since the conflict began in February 2011.
The UN children’s agency says 1,600 were killed last week alone, also the highest figure for the entire revolt.
Adrian Edwards of UNHCR discusses the refugee crisis
The Syrian Observatory for Human Rights and the Local Coordination Committees, two opposition groups, have raised their total death toll to between 23,000 and 26,000.
“President Assad assured the president of the International Committee of the Red Cross that he welcomed the humanitarian operations carried out by the committee on the ground in Syria, as long as it remains impartial and independent,” state news agency SANA said.
Assad told Maurer on Tuesday that he supports the work of the aid group in Syria so long as it remains “impartial and independent”, state television reported.
It said the ICRC chief welcomed “the co-operation shown by the Syrian government” in his meeting with Assad and “the bridges of trust that have been built between the two parties.”
Maurer was in Damascus on Tuesday for talks with Assad and other officials on ways to deliver humanitarian aid to Syrians caught up in the civil war, an ICRC spokeswoman said.
Five people were said to have been killed in the northern province on Tuesday, a day after at least 25 deaths were reported in air raids 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.