Elenzi was wanted on a European arrest warrant issued by France for allegedly having provided arms to the attacker, Mohamed Lahouaiej Bouhlel.
A Paris court in March made a formal request for 8 suspects to face trial next year.
Bouhlel was killed by police after plowing a truck down Nice’s Promenade des Anglais and into a crowd assembled for a Bastille Day fireworks display. The Islamic State claimed responsibility for the attack.