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The move is related to a complex operation to maintain the state's controlling stake in the company after Rome said last year it would sell 3 percent of Eni in 2014 to help rein in the country's massive public debt, the fourth-largest in the world.
The state Treasury holds 4.34 percent of Eni and state lender Cassa Depositi e Prestiti holds another 25.76 percent, bringing the state's total holding to 30.1 percent.
If that stake grew, stock exchange rules would force the government to make a takeover bid for the company, while a smaller holding would bring the government lower dividends.
Eni said on Friday it currently owns 0.31 percent of its own shares.