Haftar's forces claimed they destroyed Turkish drone
"Our fighter (jets) targeted and destroyed a Turkish Bayraktar aircraft as it was taking off" from Mitiga airport, Haftar's self-style Libyan National Army said in a statement on Facebook.
"The aircraft had been prepared to target our armed forces' positions," it said.
Airport authorities said they had suspended flights "until further notice" following an air strike.
An airport official, speaking on condition of anonymity, said a raid had targeted a runway without causing casualties or damage.
Haftar's forces, which hold much of eastern and southern Libya, launched an offensive in early April to seize the capital from the internationally recognised unity government.
Both sides have accused each other of receiving backing from foreign powers.
After Haftar lost a key town to unity government forces in an operation he accused Ankara of backing, he ordered his forces to target Turkish companies, ban flights and arrest Turkish nationals in Libya, his spokesman said Friday.
Turkey said Sunday six of its nationals were being held by Haftar's LNA.
Ankara has warned that it is ready to retaliate against attacks.
Libya was plunged into a bloody power struggle between multiple militias following an uprising that led to dictator Moamer Kadhafi's death in 2011.
Haftar was a retired general who had taken part in the revolt against Ghadaffi, but in May 2014 he launched his assault to purge the country of Islamist "terrorists".