The government had aimed to complete the part-privatisation of the successor to Alitalia over the summer, but then asked the two rival consortia to review their initial offers as it deemed they did not meet its goals.
The offer, whose details were not publicly disclosed, would leave the Treasury with "at least" a 40% stake and the right to appoint the company's chairman and exercise a veto on certain "strategic choices", a source has told Reuters.
The Certares-led bid was facing a rival bid presented by shipping group MSC and Germany's Lufthansa, which a source said was offering between 850 million euros ($847 million) and 900 million euros for an 80% stake.
MSC declined to comment while Lufthansa could not immediately be reached for a comment.
Prime Minister Mario Draghi's government wants to sign a preliminary agreement with the preferred bidder by Sept. 10, sources have said.
However, any deal would probably close in the final quarter of the year by which time Italy should have a new government after a national election on Sept. 25, potentially complicating the issue further.
The Brothers of Italy party, which is topping the polls, has called for the next administration to make a decision on ITA.
(Reporting by Giuseppe Fonte and Francesco Zecchini, editing by Agnieszka Flak)
