AMMAN (Reuters) - Western-backed Syrian rebels holding a strategic swathe of the desert southeast stretching to the Iraqi border said they came under major attack from government forces and allied Iranian-backed militias backed by Russian air power.

They said hundreds of troops with dozens of armored vehicles including tanks had surged into the Bir Qassab some 75 km (45 miles) southeast of Damascus towards the Badia region that skirts the borders with Jordan and Iraq in an assault that began on Tuesday.

Government forces and their allies on Wednesday took control of the Bir Qassab area, state television quoted a military official as saying. One rebel source said there had been advances by the government side.

Bir Qassab straddles the route to the eastern suburbs of Damascus, near the Dumeir air base, and is a key rebel supply line towards areas they control further southeast.

Bir Qassab fell to Free Syrian Army (FSA) rebels after it was abandoned a few months ago by Islamic State (IS) militants beating an eastward retreat to reinforce their urban bastion, Raqqa, against a U.S.-backed coalition offensive, and oil rich Deir al Zor province, which borders Iraq.

Bir Qassab, a rugged, desolate area, was a former stronghold that had given Islamic State a springboard for attacks on state-controlled territory just to the east of Damascus and a base for maintaining their grip on large swathes of the Badia region.

“The (Syrian) regime and militia ground attack started this dawn and our forces are holding on to their positions,” said Saad al Haj, spokesman for Osoud al Sharqiya, one of the largest rebel groups operating in the area.

“With (the help of) intensive Russian bombing they are trying to advance but we are repelling them,” al-Haj added.

President Bashar al-Assad’s forces, aided by Iranian-backed militias, have engaged in a race with FSA rebels in recent weeks to seize areas in the southeastern desert vacated by retreating Islamic State insurgents.

“Backed by Iran, Iraqi Shi’ite groups are spreading across the Badia,” said Abu Yaqoub, a commander in the Ahmed Abdo Martyrs Brigade, an FSA rebel group whose fighters were involved in the battles.

The army and Russian planes had been waging a relentless bombing campaign on Western-vetted rebel hideouts in the region and making steady gains, taking over several areas including the strategic Tal Dakwa mountain range, northeast of Bir Qassab.

The government offensive is part of a major campaign to recapture full control of the vast Syrian Badia, where the advance of the army and Iranian-backed militias southeast of the ancient central town of Palmyra has brought them within reach of the Iraqi frontier for the first time in years.

This move has also effectively encircled FSA-controlled desert territory stretching to the Jordanian and Iraqi borders and close to the Tanf garrison where U.S. forces are based.

Iranian-backed forces have also been trying to advance towards the base even after repeatedly being bombed by the U.S.-led coalition.

U.S. forces said on Tuesday they had shot down an armed “pro-Syrian regime” drone near the garrison. A Western intelligence source identified the drone as Iranian.

The coalition’s presence in Tanf, on the Damascus-Baghdad highway, was meant to stop Iranian-backed groups from opening an overland route between Iraq and Syria, intelligence sources have said.