0644 GMT May 23, 2019
The United Nations called for an urgent two-hour truce for the southern outskirts of the capital to allow evacuations of civilians and the wounded.
Rivalries between the two camps threaten to plunge the country into a full-blown civil war after Haftar on Thursday launched an offensive on Tripoli.
Oil-rich Libya has been driven by chaos since the uprising in 2011 that killed former leader Muammar Gaddafi, with rival administrations and armed groups vying for power.
Repeated attempts to find a peaceful solution have failed.
After a pause overnight, fierce fighting flared anew on Sunday morning south of the capital between Haftar's self-styled Libyan National Army (LNA) and forces backing the internationally recognized Government of National Accord (GNA).
As clashes raged in the mainly farmland region of Wadi Raba and the disused international airport south of the capital, a spokesman for pro-GNA forces said a "counteroffensive" had been launched to push back Haftar's forces.
Colonel Mohamed Gnounou told reporters that operation "Volcano of Anger" was aimed at "purging all Libyan cities of aggressor and illegitimate forces", in reference to Haftar forces.
The LNA meanwhile announced it had carried out its first air raid on a Tripoli suburb, despite calls by the international community to halt hostilities.
GNA head Fayez al-Sarraj warned on Saturday of a "war without a winner" and said that reinforcements were pouring into Tripoli from several regions.
Powerful armed groups from the western city of Misrata and fighters from Zentan and Zawiya – all battle-hardened militiamen who took part in the 2011 uprising that toppled Gaddafi – have joined the battle.
At least one armed group from Misrata known as Brigade 166 arrived Saturday in eastern Tripoli with dozens of vehicles, some mounted with antiaircraft guns, to join the counteroffensive, an AFP photographer said.
"We are waiting for orders to repel any advance by the enemy toward Tripoli," said the group's spokesman Khaled Abu Jazia.
Misrata forces ousted the Daesh terrorist group from Libya's coastal city of Sirte, Gaddafi's hometown, in 2016 after months of bitter fighting.
Sunday's LNA air raid came a day after forces backing the GNA launched airstrikes on their rivals for the first time since the offensive began, with at least one targeting an LNA position south of Tripoli.
LNA spokesman Ahmed Mesmari had vowed the force would retaliate.
On Saturday he said pro-Haftar forces were "progressing on several fronts... as planned" and that Tripoli would soon be captured.
Analysts say Haftar has been buoyed by a series of successful military operations that have brought all of the east and much of the south of Libya under his control.
As the tension mounted, Tripoli residents were seen Saturday queueing outside petrol stations and supermarkets.
Haftar's offensive came as UN chief Antonio Guterres visited Libya Thursday days ahead of a planned UN-backed conference aimed at uniting Libya's rivals and paving the way for elections.
The UN Security Council has called on Haftar's forces to halt their advance, warning it was putting Libya's shaky stability at further risk.
But UN Envoy Ghassan Salame insisted on Saturday the April 14-16 conference would go ahead.