"The airport was attacked this morning with mortar rounds, rockets and tank fire. It was the most intense bombardment so far," airport security official Al-Jilani Al-Dahesh said.
Islamist militia first attacked the airport last Sunday, forcing a halt to all flights.
Their continued offensive has caused extensive damage to planes and airport infrastructure that aviation officials say will keep it closed for months.
Pictures posted on social media showed a Libyan Airlines plane on fire as plumes of smoke billowed over the airport.
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He had no immediate word on any casualties.
Residents of the Qasr Bin Gheshir district close to the airport spoke of heavy exchanges with both sides deploying tanks.
The Islamists have been reinforced by other militia from Libya's third city Misrata, one of the key battlefields of the NATO-backed uprising of 2011 that toppled and killed veteran dictator Moamer Kadhafi.