Haftar's forces have advanced in mostly low-populated areas, but taking control of Tripoli constitutes a much greater challenge for them.
"I don't care who wins or loses, I just want to survive with my family", said a teacher in Tripoli, who hoped to get out.
An atmosphere of calm prevailed in the centre of the airport, while the Government of National Accord deployed additional military enforcements towards the fighting axes in southern Tripoli, coming from the cities of Zawiya and Misurata.
Libya is now split by two rival factions, with one supporting Haftar in the east, and another based in Tripoli in the west including a UN-backed government.
The oil-rich country, which has been in turmoil since the NATO-backed removal of longtime ruler Muammar Gaddafi in 2011, has two rival governments: an internationally recognised government based in Tripoli and headed by Prime Minister Fayez al-Sarraj, and an administration based in the eastern city of Tobruk, which is allied with Haftar.
After a pause overnight, fighting resumed Monday morning around the capital's destroyed main airport, some 30 kilometers (18 miles) south of Tripoli, and the rural area of Wadi Rabi further east.
Meanwhile, parliament Speaker Aguila Saleh said that the LNA's march on Tripoli is in line with the constitutional declaration and parliament decision to rid the capital of militias. Haftar is a tough anti-Islamist who has the support of Egypt and the UAE and is strong in eastern Libya.
But the Kremlin on Monday urged "all sides to reject actions that could provoke bloodshed in battle and the deaths of civilians".
Serraj's forces carried out an air strike on an LNA position in the suburb of Suq al-Khamis on Tuesday, a resident and an eastern military source said, without giving more details.
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In Tripoli, Higher Council of State chief Khaled al-Mishri announced his rejection of the LNA's operation, calling on the GNA to issue a warrant for Haftar's arrest.
Spokesman Colonel Mohamed Gnounou said it was aimed at "purging all Libyan cities of aggressor and illegitimate forces", in reference to Haftar's fighters.
The previous day, fighting raged throughout a two-hour window set by the United Nations for a pause in hostilities for civilians and the wounded to flee.
A spokesman for the Tripoli-based Health Ministry said fighting in the south of the capital had killed at least 25 people, including fighters and civilians, and wounded 80.
Hifter's forces said Saturday they had seized the old airport.
The United Nations said 2,800 people had been displaced by the clashes and many more could flee, though some were trapped.
It was launched just days ahead of a planned United Nations conference aimed at uniting Libya's rivals and paving the way for elections.
The UN had scheduled a three-day conference on April 14 in the southwestern town of Ghadames to discuss a constitutional framework for elections as a way to end the North African country's eight-year crisis.