Hezbollah fighters and Syrian soldiers were “tightening the noose” on ISIS in the barren frontier region, the military media unit run by Hezbollah said.
Fear of conscription has been a major sticking point in the local agreements, encouraging residents to leave for rebel-held areas of northern Syria.
The seized oil fields lie south of Rasafa and its oil wells, which the army took last month in their first major territorial gains, inside the province.
The Syrian army said it would suspend combat operations in southern Syria until Thursday but rebels said the army had already violated the ceasefire.
ISIS is on the back foot in Syria after losing territory in the north to an alliance of US-backed, Kurdish-led militias and to Turkey-backed Syrian rebel groups.
ISIS has captured Palmyra, whose ancient ruins are a UNESCO World Heritage Site, twice during Syria’s six-year conflict, last in December 2016.
The fate of those stuck in the last rebel bastion in Aleppo is still at stake after a series of sudden advances by the Syrian army and allied Shi’ite militias under an intense bombardment that pulverised large sections of the city.
Recapturing the entire rebel pocket of Aleppo will constitute the biggest battlefield victory yet for Bashar al-Assad and his military coalition of Russia’s air force, Iran and Shi’ite militias.
The options facing those caught up in the rapid government advance are bleak: men of fighting age could be arrested whether they stay put or head to government-held districts. If they flee to rapidly-shrinking rebel-held areas they may only be putting that prospect off.
The rebels, who controlled large parts of eastern Aleppo for nearly five years, have lost around two-thirds of their territory in the city over the past two weeks.
Syria’s army has said that anybody who remains in the city of Aleppo after offering those who wish to leave an opportunity to do so would face their “inevitable fate”.
The Syrian Observatory for Human Rights has observed a significant drop in the violence in areas covered by the truce.
The assault by the rebels is by far the biggest military campaign waged by the insurgents against the government forces in recent months.