Jordan has kept its border firmly sealed, insisting it can not handle any more than the 650,000 registered Syrian refugees it already houses, and has instead sent aid into its war-torn neighbour.
The Britain-based Syrian Observatory for Human Rights said Monday's fighting was concentrated near the villages of Tafas and Nawa.
Jordan's foreign minister said on Monday that he will hold talks with his Russian counterpart over the conflict in Syria, where the latest government offensive in the country's south has displaced 270,000 people, according to a United Nations official.
A girl holds a Syrian flag and a picture of Syrian President Bashar al-Assad in Deraa, Syria July 4, 2018.
Syrian rebel negotiators began a new round of talks with Russian officers on Tuesday over a peace deal in southern Syria under which they would hand over weapons and allow Russian military police to enter rebel-held towns.
"Negotiations with the Russian enemy in Busra al-Sham have failed, after they insisted on the surrender of heavy weapons", it said in an online statement.
The Russian government has backed the Syrian regime's heavy military campaign to recover the area.
The rebels said the talks had failed on Saturday afternoon but the process has since been revived after Jordan stepped in.
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The state media said the military operations are going in tandem with the reconciliation efforts, which explains the situation where several towns accepted the entry of the Syrian army without battles.
It included a ceasefire, the army's withdrawal from towns it had already taken, and safe passage to other opposition territory for rebels or civilians who did not want to live under regime control.
"Intense air and ground based strikes have reportedly continued in multiple areas in Syria's Daraa Governorate, resulting in the death and injury of civilians and the largest displacement in the area since the conflict began", United Nations deputy spokesman Farhan Haq told reporters.
Air raids meanwhile intensified, said the monitor, the Syrian Observatory for Human Rights, as displaced people flocked to the border areas least likely to be hit.
Both countries have kept their borders closed to those fleeing the violence, despite mounting calls by rights groups to let Syrians escape to safety.
"The abject refusal by Jordanian authorities to allow asylum seekers to seek protection not only goes against their global legal obligations, but against basic human decency", Lama Fakih, deputy Middle East director at Human Rights Watch, said.
Safadi said Amman keeps open channels with Damascus and Moscow and that his talks with Lavrov will focus on reaching a cease-fire and halting the displacement.