The UN said at least 2,500 people had been displaced in Deraa province in recent days but a monitoring group put the figure at about 12,500.
Troops and pro-government militiamen have stepped up artillery strikes on the region, which borders Jordan and the Israeli-occupied Golan Heights.
Fighting has been limited there since last year under a “de-escalation” deal.
The US – which supports rebel forces in the area and brokered the agreement along with Jordan and the government’s key ally, Russia – warned last week that it would “take firm and appropriate measures” in response to any violations.
The Syrian Observatory for Human Rights, a UK-based monitoring group, reported on Thursday that some 12,500 civilians had fled their homes in eastern Deraa province in the past two days after government forces intensified their shelling of the area, killing at least eight civilians and six rebel fighters.
The displaced civilians were heading southwards to rebel-held areas near the Jordanian border that had not yet been affected by the bombardment, it said.Paul Craig Roberts: “The entire western world lives in cognitive dissonance”