Ahmed Mismari, a spokesman for Haftar's Libyan National Army (LNA), described a new facet to the clashes, saying: "Air forces took part for the first time in the military operations. It conducted a very successful operation to secure the airport road (to the city center)."
Colonel Mohamed Gnounou, a spokesman for pro-government forces, said its counteroffensive against Haftar aimed at "purging all Libyan cities of aggressor and illegitimate forces."
The EU's foreign policy chief, Federica Mogherini, called on leaders to avoid further military escalation and urged a "return to the political process." She added that developments in Libya were "definitely not going in the right direction."
Peter Millet, the UK's former ambassador to Libya, told DW international involvement should refrain from escalating the situation, saying: "I don't think the international community can, could or should intervene militarily in Libya."
Under a UN-backed deal last year, the Tripoli-based government of Fayez al-Sarraj, Haftar and some warring militias agreed to a transition deal that included holding national elections by the end of 2019.
But Haftar became frustrated with the Tripoli government's inability to govern effectively. Last week, Haftar ordered his forces to seize control of the Libyan capital.
The UN-backed government accused Haftar of betraying the transition deal.
Who is Khalifa Haftar? Haftar is a former military officer who helped Gadhafi during a military-led coup in 1969. However, he later fell out of favor with the dictator. He enjoys widespread support in eastern Libya from several armed militias filled with battle-hardened fighters who made their names during the 2011 civil conflict.