At least 270,000 people have fled their homes in south-western Syria since the military launched an assault on rebel-held areas two weeks ago, the UN says.
Many of those displaced by the fighting in Deraa and Quneitra provinces have headed towards the borders with Jordan and the Israeli-occupied Golan Heights.
But neither country has said it will allow an influx of refugees, sparking fears of a humanitarian crisis.
Government forces have been advancing with the help of Russian air strikes.
On Sunday, rebels in the major town of Bosra al-Sham reportedly agreed to lay down their arms and accept President Bashar al-Assad’s rule.
More than 130 civilians have been killed since hostilities escalated, according to the Syrian Observatory for Human Rights, a UK-based monitoring group.
Deraa and Quneitra had been relatively calm for almost a year because of a “de-escalation” agreement brokered by the US and Jordan, which support the opposition, and Russia, a staunch ally of the government.
But Mr Assad set his sights on regaining full control of the provinces after defeating rebels in the Eastern Ghouta region outside the capital Damascus in April.