The attack took place in Sidi Khalifa district east of Benghazi, Egaim told Libya’s Nebe TV channel.
He said he was unhurt but four of his guards were wounded.
Salah Bulghaib, chief of military intelligence, accused Khalifa Haftar, a military commander for Libya’s Tobruk-based government of being behind the assassination attempt.
Bulghaib claimed that according to intelligence reports they had received, the attack targeted him, Egaim and the heads of Egaim’s Al-Awaqir tribe.
The attempt was a retaliation for the arrest of one of the perpetrators of the al-Abyar massacre, Bulghaib said.
In October, a mass grave with 37 bodies — handcuffed and shot in the head — was discovered in al-Abyar, a district in Benghazi.
Libya has been dogged by violence since 2011, when a bloody uprising led to the ouster and death of longstanding President Muammar Gaddafi after more than four decades in power.
The ensuing power vacuum led to the emergence of several rival seats of government and a plethora of heavily-armed militia groups.