Pictures on social media purportedly showed Misrata fighters celebrating at the terminal building and standing on civilian planes in what, if confirmed, would be a big development in the battle to control the capital.
Arab channels Jazeera and Al-Arabiya also said Misrata forces were controlling the airport, although Reuters was unable immediately to access the area. Heavy shelling could be heard in other parts of the city.
War planes had earlier struck Misrata positions in Tripoli in an attack claimed by renegade general Khalifa Haftar. The raids killed 10 people and wounded dozens, the Misrata faction said.
The fighting is the worst since the overthrow of Muammar Gaddafi in 2011.
In the Nato-backed campaign to oust Gaddafi, fighters from the western region of Zintan and Misrata, east of Tripoli, were comrades-in-arms. But they later fell out and this year, have turned parts of Tripoli into a battlefield.
Haftar launched a campaign against Islamists in the eastern city of Benghazi in May and threw his weight behind the Zintan fighters.
In Saturday’s fighting, residents heard explosions early in the morning near the airport, where the two groups have been fighting for control for more than a month.
Local television channel al-Nabaa said planes had attacked four Misrata positions. A Misrata spokesman said the planes had come from Egypt and the United Arab Emirates, two countries which have cracked down on Islamists.
But Haftar’s air defence commander, Sager al-Jouroushi, said that his forces were responsible for the attack. Haftar’s forces also claimed responsibility for air raids on Misrata positions in Tripoli on Monday.
Western countries and Egypt, worried about Libya becoming a failed state and safe haven for Islamist militants, have denied any involvement. The Libyan government has said it does not know who is responsible for the air attacks.
In a challenge to the parliament elected on June 25, the spokesman for Operation Dawn called for the old General National Congress (GNC) to be reinstated. Misrata forces have rejected the new House of Representatives, where liberals and lawmakers campaigning for a federalist system have made a strong showing.
In a sign of deep divisions between Libya’s regions and political factions the House of Representative declared the Operation Dawn as well as militant Islamists like the Ansar al-Sharia as “terrorist groups”.
“This is a war between the Libyan state and the state institutions led by our sons, soldiers and officers in the army, against terrorist groups outside of the law,” the house said in a statement.
Fresh fighting also erupted between Haftar’s troops and allied army special forces with Islamists in two Benghazi suburbs, where loud explosions could be heard. Four soldiers were killed and 31 wounded, a hospital medic said.
Prime Minister Abdullah al-Thani called on Egypt and Tunisia to open their airspace again for flights to western Libya. Both countries cancelled most flights to Libya for security reasons after the air strikes, cutting off a vital link to the outside for Libyans and foreigners fleeing fighting.
Libya has used the small Matiga airport in Tripoli for civilian traffic since the main airport was turned into a battlefield last month. The tower, runway and at least 20 aircraft have been damaged, officials have said.
When flying into Matiga, passengers can sometimes see smoke rising from battles in and around the main airport.
The violence has prompted the United Nations and foreign embassies in Libya to evacuate their staff and citizens, and foreign airlines largely stopped flying to Libya.
Tripoli has mostly slipped out of control of the government, with senior officials working from Tobruk in the east, where the new parliament has based itself to escape the violence in Tripoli and Benghazi.
Libya’s central government lacks a functioning national army and relies on militia for public security. But while these forces receive state salaries and wear uniforms, they report in practice to their own commanders and towns.