Aer Lingus has shot down a surprise bid from Ryanair to seize control, saying its offer is too low.
The former state-owned airline said it was unlikely any buy-out could be completed anyway, considering Michael O'Leary's firm was already being investigated by the UK Competition Commission over its existing 30% share.
Ryanair's fresh offer would value Aer Lingus at 694 million euro (£560 million).
"The Board, having considered the offer with its advisers, believes the offer, even if it is capable of completion, undervalues Aer Lingus," a spokesman said.
He added that Ryanair's two previous failed attempts to take over the airline suggested its newest effort will fall flat too.
"The Board notes that Ryanair's unsolicited offer in 2006 was blocked by the European Commission and was not capable of completion and that Ryanair's second offer, in 2008, was withdrawn," he went on.
"Consequently, there is significant uncertainty that any offer from Ryanair, if made, would be capable of completion."
Aer Lingus also urged shareholders to do nothing following the bid.
Earlier, Taoiseach Enda Kenny said the Government would not be forced into any fire sale of its remaining 25% share in the airline.
The Government has been hoping to sell its stake as a state asset following an agreement with its bailout masters the Troika earlier in the year.
Public expenditure minister Brendan Howlin said money raised from the sale would go towards job creation.
If Ryanair's bid is accepted, its offer of 1.30 euro per share would result in a 175 million euro (£141 million) windfall for the Exchequer.
"In consideration of any such (state) asset being disposed of, the Government will not be forced into any fire sale," Mr Kenny said.
"The Government will use proper and appropriate consideration to the right timing, to the right price and in the best interest of the country and the people."
But he added that the Government could not block or veto a takeover.
Ryanair chief Mr O'Leary hopes to boost Aer Lingus's passenger numbers by 4.5 million to 14 million over the next five years by forming "one strong Irish airline" to compete with major European players.
A previous takeover attempt in 2006 was rejected by regulators but Ryanair said consolidation and the economic downturn had left Aer Lingus "exposed as a small and uncompetitive airline".
Opposition party Fianna Fail has appealed to the Government to use its stake in Aer Lingus to block the buy-out.
Transport spokesman Timmy Dooley said a consolidation of the two airlines would result in reduced competition, increased fares and less choice.
Sinn Fein public expenditure spokeswoman Mary Lou McDonald made a similar plea to the Government to block the sale, saying the airline was of "strategic importance" to the State and was vital to maintain the island's connectivity with the rest of the world.
The Irish Hoteliers Federation also called on the Government to try to prevent the Ryanair takeover.
Federation president Michael Vaughan said even if Mr O'Leary operated both airlines separately, he would still maintain a monopoly which would be damaging.
"As it stands, Ireland has already become overly reliant on Ryanair and Aer Lingus, which between them now account for almost 80% of traffic into and out of Dublin," said Mr Vaughan.
"They are also the dominant players across Ireland's regional airports.
"A takeover of Aer Lingus by Ryanair would only exacerbate the situation and would not be aligned with the interests of the Irish economy."