On 1 June staff read out statements condemning the abduction of the Mellitah employee Yousef al-Shoumani on Libyas al-Nabaa television channel and called for better security for oil workers.
It is unclear who is behind the kidnapping.
Eni is the only foreign oil company that has managed to maintain production levels amid the countrys worsening conflict.
Speaking in a conference call on 29 April after the company published its first-quarter results, chief upstream officer Antonio Vella said Eni is currently producing about 300,000 barrels a day (b/d) in Libya.
Before the 2011 uprising, Eni was producing about 280,000 b/d.