According to the report, the plane was shot down in the area of Al-Hir, South of Tripoli.
The LNA’s forces have previously destroyed GNA aircraft, as well as planes belonging to group allied to it.
The pilot from a Mirage F.1ED aircraft has been captured by the LNA’s forces, the report added.
The incident comes a day after Haftar reportedly ordered the LNA forces to chase and destroy enemy troops.
On 4 April, the Libyan National Army launched an offensive on Tripoli, which is controlled by the UN-backed Government of National Accord. Two days later the GNA announced a counteroffensive dubbed “Volcano of Rage”.
More than 430 people have been killed and more than 2,000 have been injured since the beginning of the offensive on Tripoli, according to the World Health Organisation.
Since the overthrow and killing of Libya’s long-time leader Muammar Gaddafi in 2011 the coutry has been divided into two parts, the internationally recognised and GNA headed by Fayez Sarraj in Tripoli and a parallel administration allied to Haftar in the East of the country.