Air raids continued in the meantime, said the Syrian Observatory for Human Rights, in an offensive that the United Nations says has driven 160,000 people from their homes, threatening a humanitarian catastrophe.
Zeid, a Jordanian prince, said there is a grave risk that the intensified fighting will see many civilians trapped, caught between government forces on the one hand and rebels and members of the Islamic State group on the other.
The onslaught has sparked fears of a re-run of the offensives a year ago against the rebel enclaves of Aleppo and eastern Ghouta, including deadly bombardments followed by a retaking of territory and an accord to evacuate rebels from the areas.
The current pro-government push to retake southwestern Syria is the largest assault on opposition-held territory there since a de-escalation deal brokered by the United States, Russia and Jordan came into effect in July 2017.
According to a war monitoring group, the offensive has uprooted more than 120,000 civilians, driving majority towards the border with Jordan and the Israeli-occupied Golan Heights. Another rebel official said some towns were trying to negotiate deals with the state on their own.
Late on Saturday, the Jordanian government said the army had started delivering humanitarian aid to thousands who had taken shelter across the frontier in rebel-held Syria.
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.
The regime has chipped away at rebel-held territory in Daraa since the escalation began nearly two weeks ago.
Syria is pushing south in an offensive against rebels in the southern Daraa Province, which borders Israel and Jordan.
Mr Safadi also said he would hold talks with his Russian counterpart in Moscow on Tuesday, which he hoped would "take us more steps forward to contain this crisis and prevent more destruction".
Almost 100 civilians have been killed since the assault began on June 19 with airstrikes, rocket fire and crude barrel bombs.