Syrian regime troops advance toward Jordan border crossing
Witnesses also told Reuters they spotted a Russian tank and armoured vehicles heading toward the border post.
Syria's cash-strapped government hopes to recapture Nassib so that it can reopen trade with Jordan to the south.
The news follows Thursday's announcement that regime forces seized their first security post along Jordan's border in more than three years, the Syrian Observatory for Human Rights reported at the time.
Regime ally Moscow has, since intervening militarily in Syria in 2015, employed a carrot-and-stick approach that has allowed President Bashar al-Assad's forces to recapture significant territory across the country.
Under various ceasefire deals, rebels hand over heavy weapons, local police take control of the area and state institutions resume working there.
More than 30 rebel towns in Daraa have agreed to fall back under regime control through these agreements, doubling Assad's hold on the province to around two-thirds.
However, rebels still hold Daraa's western countryside, the southern half of the divided provincial capital and the prized Nassib border crossing.
"After the town of Al-Mutaaiyah agreed to the regime taking over, government forces are now four kilometres (2.5 miles) east of the crossing," said Rami Abdel Rahman, head of the Britain-based Observatory.
Daraa is considered the cradle of the 2011 uprising against Assad that triggered Syria's war.
Rebel territory in the south was included in a ceasefire brokered last year by Russia, the United States and Jordan. But the regime along with ally Moscow has violated the deal.
On Thursday, the UN Security Council convened in an urgent closed-door meeting to discuss the south, but Russia blocked the council from adopting a statement.
More than 150 civilians have been killed since the start of the two-week-old Daraa offensive.
The offensive has also pushed more than 320,000 people to flee, according to the UN, many to the closed border with Jordan or west near the Israeli-occupied Golan.
Agencies contributed to this report.