The fighters of the Shura Revolutionary Council, which includes the Islamist Ansar al-Sharia group, have been trying since early September to capture the facility, which houses both civilian and military airfields.
Oil-rich Libya slid into chaos after Moamer Kadhafi was toppled in a NATO-backed uprising three years ago, with interim authorities confronting powerful militias that fought to oust the veteran dictator.
Today, the AFP correspondent in Benghazi said heavy artillery fire could be heard since dawn in the southern suburb of Benina, where the airport is located.
In May, Haftar launched a campaign to wipe out Islamists in Libya.
There has been almost daily fighting between his forces and the militias, which now control nearly all of Benghazi, Libya's second city and the cradle of the 2011 uprising.
Meanwhile, a senior militia commander and former MP has been killed in fighting with rival militias west of the capital Tripoli, one of his comrades and a medical source said.
Another source, from Kilani's hometown of Zawiyah, said he had been killed in an ambush along with three companions as they were returning to Tripoli.
A hospital source in Zawiyah confirmed receiving Kilani's body, along with three others.
Kilani had served in the previous parliament as a member of the Wafa bloc, accused by its rivals of backing radical Islamists in Libya.
Its fighters, a powerful presence in the capital, called the same month for the reinstatement of the outgoing General National Congress.
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