ENI is the biggest foreign oil company in Libya and runs a pipeline to Sicily from the Mellitah gas terminal in the troubled country.
"What we are worried about at the moment is the Mellitah terminal, which has been attacked by protesters, pushing us to stop exports towards Italy," Italian media quoted chief executive Paolo Scaroni as saying.
However, Scaroni said he did not forsee problems with gas supply to Italy.
The ENI press office was unable to say whether the attackers were armed.
Last week, protesters from Libya's minority Amazigh Berber ethnic group held a sit-in at the Mellitah terminal to demand greater rights.
Berbers make up about 10 percent of Libya's population. They were persecuted under former dictator Muammar Qaddafi and feel marginalized under the new regime even though they played a key role in the 2011 uprising.
The company that manages the Mellitah terminal is owned jointly by ENI and Libya's National Oil Corporation (NOC). It supplies Italy with 17 million cubic metres of gas a day.