Meanwhile, 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, and the United Nations warned of a humanitarian catastrophe.
A war monitoring group said the offensive has uprooted more than 120,000 civilians in the southwest since it began last week.
Assad said earlier this month that he was still pursuing a political solution for Syria's rebel-held southwest, but would use military force if the effort failed.
Russian negotiators have demanded rebels accept terms like those agreed for eastern Ghouta, where insurgents either left for opposition territory in the northwest along with their families or accepted the return of state rule, Jabawi said.
The Syrian Observatory for Human Rights said on Saturday that at least eight battered rebel-held towns in Daraa returned to regime control under Russian-brokered deals.
"The IDF will also insist on the principle of non-involvement in the situation in Syria, alongside a policy of a firm response to any violation of the State of Israel's sovereignty and creating a risk to its residents", the IDF stated.
These include around 20,000 to areas near the Naseeb border crossing with Jordan, a country that already hosts more than 650 000 registered Syrian refugees and says the actual figure is closer to 1.3 million.
Southwest Syria is a "de-escalation zone" of reduced warfare and bombardment agreed by Russia, Jordan and the United States a year ago.
"Free Syrian Army (FSA) has been asked to hand down all of the heavy and light weapons it has".
"[He] calls on all parties to respect their obligations under global humanitarian law and human rights law, protect civilians and facilitate safe, unimpeded and sustained humanitarian access".
A rebel-held town east of the city of Daraa during airstrikes on Saturday.
The UN refugee agency has reported that the number of displaced civilians in southern Syria has more than tripled in recent days, with over 160,000 now displaced from their homes in a little over a week.
Still formally at war with Syria, Israel has waded into the conflict only sporadically, either to curtail Iran's growing influence in the region or when there is spillover from battles taking place along its border.
Syria's southwest is strategically sensitive because of its proximity to the frontiers with both Jordan and the Israeli-occupied Golan Heights.
It is estimated that as many as 115 civilians have been killed in regime bombardments on the province since 19 June.
In the Syrian province of Deraa in the south-west of the country, Syria's government army continues the operation against groups of terrorists.