The New Syrian Army said the operation -- launched on Tuesday -- is aimed at severing the supply line through the Albu Kamal crossing that links areas under IS control in eastern Syria and Iraq 's Anbar province.
The rebels said the operation was coordinated with Iraqi forces who were advancing on the crossing from the other side of the border.
IS seized the crossing in mid-2014, when it overran swathes of territory on both sides of the border and declared a self-styled "caliphate".
The Syrian Observatory for Human Rights said the NSA had advanced to within five kilometres (three miles) of Albu Kamal from both the northwest and southwest.
As NSA forces edged closer, IS beheaded five young men in the town it said were working with the US-backed group, the Britain-based monitoring group said.
"The NSA and supporting forces captured Al-Hamdan airport, five kilometres northwest of Albu Kamal with air support from the coalition," said Observatory chief Rami Abdel Rahman.
He had no immediate word on casualties in the fighting.