As of Saturday, government forces controlled more than half of Daraa province, up from just 30 percent at the start of the operation, the Syrian Observatory for Human Rights monitor said.
Russian Federation has played a critical role in supporting Assad's two-week offensive with air power and negotiating local deals initially overseen by its military police.
It would also allow the regime to reopen the Naseeb border crossing with neighboring Jordan and resume lucrative trade across the frontier.
Southwest Syria is a "de-escalation zone" of reduced warfare and bombardment agreed by Russia, Jordan and the United States previous year. "These are people who fled their homes with nothing. we realized we had to do something different".
President Bashar al-Assad has pressed ahead with the offensive despite US warnings of "serious repercussions".
Free Syrian Army rebels are resisting the plan, proposing instead that Russian military police take control rather than Syrian military forces.
Bosra al-Sham, whose black-rock Roman citadel and theater are a UNESCO World Heritage site, was captured by rebels in 2015.
"The (rebels') negotiating team refused to surrender and refused to accept the Russian conditions".
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Jordan's Foreign Minister Ayman Safadi said the kingdom was engaged in intensive diplomacy with all parties in the conflict to help broker a ceasefire that would ease the plight of displaced civilians.
On Saturday at least 10 civilians were killed when bombs were dropped on the rebel-held village of Ghasam, relief workers said.
At least five at people were killed in regime bombardment of Deraa on Saturday, the Observatory said, taking the civilian death toll in the province to 105, including 19 children, since June 19.
Syria's southwest is strategically sensitive because of its proximity to the frontiers with both Jordan and the Israeli-occupied Golan Heights.
In Washington, a State Department official said the United States could not confirm the truce report, and the situation in southern Syria remained "grim" with Russian Federation and Syrian government forces continuing to bomb the area.
A ceasefire has been agreed for southern Syria between the government and rebels, a Jordanian official source said on June 29, amid fears of a gathering humanitarian catastrophe in a region sensitive to neighbours Jordan and Israel.
Israel has repeatedly warned that it will not allow Iran or Iranian-backed forces to entrench militarily in the country, where they have for years provided military backing to Assad's government forces.
An Israeli army commander told Reuters it was hard to quantify how many people had sought shelter in the area immediately across the border, but that it was in the thousands and there were hundreds more arriving each day.
Retaking the whole of Daraa province would be a symbolic victory for the regime, as it is seen as the cradle of the anti-Assad uprising seven years ago that spiraled into civil war.