State media made no mention of the agreement.
This Saturday, June 30, 2018 photo provided by Nabaa Media, a Syrian opposition media outlet, shows people who fled from Daraa, walk outside their tents near the Syria-Israel border, southern Syria.
But Abazeed said the evacuation of opposition fighters and their families planned for Sunday was temporarily put on hold due to "an exchange of fire" between rebels and regime forces.
Though Jordan was an important sponsor of rebels in the south, analysts said this support was always aimed primarily at safeguarding its border from security threats such as Islamic State militants, not toppling Assad.
Neighboring Quneitra province remains under rebel control for now despite also being the target of recent airstrikes.
Jordan, which is already hosting some 650,000 Syrian refugees, has refused so far, citing a combination of security and economic pressures, the Reuters news agency said.
These concessions will potentially allow tens of thousands of displaced Syrians along the Jordanian border to return to their homes as many now fear being targeted by government forces.
Under the deal, Russian military police will take the border crossings, and will guarantee the return of displaced civilians and rebels who want to end their opposition to government rule. The Syrian flag flew in the distance, along with what appeared to be a Russian flag.
Ibrahim Jabbawi, another spokesman for the rebels' southern operations, confirmed the agreement on a ceasefire, the surrender of heavy weapons, and the rebel handover of the frontier road with Jordan.
The region had been captured by rebels in 2015.
Most of the hospitals had shut down amid the destruction in insurgent territory, which now barely had access to water or electricity, he said.
Anders Pedersen, the United Nations resident and humanitarian coordinator in Jordan said on Sunday that only about 150-200 people remained there.
The initial phases would cover the area along the border with Jordan, rather than the parts of north-western Deraa around the city of Nawa, he said.
Commenting on the fate of the terrorist organisations deployed in the south of Syria, Massaid said there was a state of chaos, but in light of the Syrian army's return, the terrorists will move into the Syrian northern territories or any border area, noting that their current number is estimated at 2,500 to 3,000.
There are only "around 150 to 200 people right now at the border", he said.
Around 60,000 have sought shelter near the border with Jordan and thousands more at the frontier with the Israeli-occupied Golan Heights.