The army lost control of the strategic motorway to rebels four years ago.
Capturing the access road to the airport will facilitate the military operation in the airport area and back the ongoing military campaign in the eastern neighbourhoods of the city.
The Syrian army and allied militia also advanced towards rebel-held areas of Aleppo's Old City, thrusting deeper into opposition parts of the city in a relentless attack which a military source said would be over in a matter of weeks.
During the past few hours, the army also took control of al Maysar, Karam and al Qaterji neighborhoods.
Restoring full control over Aleppo would mark the biggest triumph yet for the Syrian President Bashar al-Assad in a war that spiralled from protests against his rule in 2011. The campaign waged by the Syrian army and its allies in Aleppo is one of the most ferocious of the war, with hundreds reported killed in recent weeks alone.
Advance on the Old City
The Syrian army and allied militia have advanced towards rebel-held areas of Aleppo's Old City, thrusting deeper into opposition parts of the city in a relentless attack which a military source said would be over in a matter of weeks.
A senior rebel official told Reuters rebel groups in Aleppo had told the United States they will not leave their besieged, shrinking enclave, responding to Russian call for talks with Washington over their withdrawal.
But facing relentless bombardment and ground assaults, the rebels may eventually have no choice but to negotiate a withdrawal from eastern Aleppo, where tens of thousands of civilians are thought to be sheltering.
The Western and regional states that have backed the rebellion appear unwilling or unable to do anything to prevent a major defeat for the opposition fighting to topple President Bashar al-Assad, whose campaign to regain all Aleppo has been backed by the Russian air force and Iranian-sponsored militias.
The army announced new advances, which were confirmed by a rebel official with the Jabha Shamiya group. The advancing forces may soon reach a strategically important eye hospital, the capture of which would threaten further gains near the Old City, the official said, speaking to Reuters from Turkey on Sunday.
Loud explosions were heard coming from eastern Aleppo as night fell, Reuters journalists in the government-held part of the city said. The Jabha Shamiya official said further advances may force a rebel withdrawal to the southwestern corner of their enclave. "The areas of Old Aleppo will be threatened to a great degree," said the official. "It is scorched earth."
The UN Security Council is due to vote on Monday on a draft resolution that would demand an initial seven-day truce in Aleppo, which could then be renewed. But it was unclear if veto-power Moscow would block the resolution.
Russia said on Saturday it was ready for talks with the US over a full withdrawal of rebels from Aleppo.
Speaking to Reuters from Turkey, senior rebel official Zakaria Malahifji of the Fastaqim group said rebels fighting in Aleppo told US officials on Saturday they would not leave the city.
"Our response to the Americans was as follows: 'we cannot leave our city, our homes, to the mercenary militias that the regime has mobilised in Aleppo'," said Malahifji.
The United States has yet to comment on the proposal made by Russian Foreign Minister Sergei Lavrov for talks on the withdrawal of all rebel fighters "without exclusion" from Aleppo.
The Syrian army has vowed to crush the rebels in Aleppo.
"The expectation is weeks," the military source said, referring to the timeframe for taking back the whole city.
Food and fuel supplies are critically low in eastern Aleppo, where hospitals have been repeatedly bombed out of operation.
