The internally displaced Syrians reportedly turned back after Israeli soldiers shouted at them to "go back", after they crossed the Alpha Line near Tel al-Haarra, a strategic hill which overlooks Israel's Golan Heights. On Sunday, the first batch of armed fighters and their families left the city of Daraa, the provincial capital, in buses that would take them to the rebel-held Idlib province in the north. They are trapped between the Israeli border and advancing regime forces with little shelter from the harsh summer weather.
On Tuesday, a monitor reported seven Syrians killed in regime and Russian bombings near the frontier.
The so-called Syrian Observatory for Human Rights (SOHR) also confirmed that Damascus government units had made a "quick advance" against militants in the western part of Dara'a.
"Buses began moving from the gathering point towards the edge of the city to be searched", said Mr Abdel Rahman.
Troops loyal to President Bashar al-Assad took control of the al-Haara hill on the second day of an offensive aimed at retaking the remaining pockets of opposition in Quneitra province along the border with Israel.
The army offensive aims at completing its control of the strategic area known as the "Triangle of Death", which connects southern the Damascus countryside with Deraa and Quneitra provinces.
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The regime now controls Daraa and its eastern outskirts, while its forces have been deployed along most of Syria's border with Jordan, in line with an earlier agreement between Russian Federation and Syrian armed opposition groups.
Meanwhile, the Syrian Arab Red Crescent said that a humanitarian aid convoy reached government-held parts of southwest Syria. Israel does not want its enemies Iran and Hezbollah, both allies of Assad, to move forces near its border.
Abdel Rahman said that unlike in Daraa, fighter planes had not been used in the Quneitra offensive and that Russian Federation was not yet taking part.
Though the objective of this demilitarized zone was to prevent further clashes between the two countries, by calling for a strict adherence to the ceasefire agreement, Israel has also established a de facto safe zone where fleeing Syrians can be spared from the Assad regime's offensive, to a certain degree.
Rebels and militant groups still control 70 percent of Quneitra, while the Syrian government is in full control of Sweida - a third southern province - the Observatory added.
Both Israel and Jordan have signaled their consent to the campaign, though Israel has said it will not accept the presence of Iranian forces or proxy forces along its frontier.