Reports say Aer Lingus is set to accept the latest takeover offer from International Airlines Group (IAG).
The parent company of British Airways (BA) has approached the Irish carrier about a fresh takeover bid.
Sources said that the board of IAG had authorised an improved all-cash offer earlier this week worth at least €2.50 a share, which would value the airline at more than €1.3bn.
Directors of Aer Lingus discussed the proposal on Friday with their investment banking advisers from Goldman Sachs, according to insiders.
But the Financial Times has now said Aer Lingus is expected to recommend the bid to its board.
"People familiar with the matter said that the two sides could announce the start of formal talks to finalise a deal in the next few days, but warned that the situation remained fluid," the paper says.
The fresh overture could be sufficient to persuade Aer Lingus to enter into formal takeover discussions with IAG, although it was unclear this weekend whether there were significant conditions attached to the proposal.
It was also unclear whether IAG might be prepared to raise its offer for a third time if the current proposal is rejected.
IAG chief executive, Willie Walsh, is a former Aer Lingus pilot who went on to run the airline before taking the helm at BA in 2005.
He has made two previous approaches for the carrier, pitched at €2.30 and €2.40 a share, in the past six weeks.
Both were rebuffed by Aer Lingus on the basis that there were undisclosed conditions attached and that they "fundamentally undervalue[d] Aer Lingus and its attractive prospects".
Mr Walsh's attempt to acquire Aer Lingus is designed to cement its grip on take-off and landing rights at London's Heathrow Airport, while enabling him to improve the Irish carrier's profitability by combining some operations with those of IAG.
Already the largest carrier at Heathrow, a merger of the two companies would create a group with close to half of the available slots there.
Ryanair and government stakes
Even if the Aer Lingus board is minded to open talks with IAG, Mr Walsh will need to persuade the Irish government and Ryanair chief executive Michael O'Leary of the bid's merits.
Ryanair owns a 29.8% stake in Aer Lingus and has fought a long-running battle with regulators over both that shareholding and a string of its own bids for its rival dating back to 2006.
Ryanair has indicated that it would be prepared to consider an offer of between €2.50 and €2.70 a share.
The Irish government holds a 25.1% stake in the airline, and reports have suggested that it could insist that IAG retains Aer Lingus's Heathrow slots solely for flights to and from Ireland as a condition for approving a deal.
Analysts have argued that such a pre-condition would make Aer Lingus less attractive to Mr Walsh, who in addition to his IAG role is also chairman of the National Treasury Management Agency (NTMA).
IAG was created in 2009 from the merger of BA and Iberia, which has been radically restructured by Mr Walsh against initially intense opposition from Spanish labour groups.
Since then, it has also acquired Vueling, another Spanish carrier, struck an alliance with American Airlines and considered several other big takeovers.
Aer Lingus is preparing for a transition in its leadership regardless of Mr Walsh's efforts to acquire it.
The airline chief executive, Christoph Mueller, is leaving in May to run Malaysia Airlines - which is being nationalised following the disasters last year involving flights MH370 and MH17.
IAG, which is being advised by Deutsche Bank, and Aer Lingus both declined to comment.
IAG commented on the proposed deal back in December, saying: "There can be no certainty that any further proposal or offer will be forthcoming. A further statement will be made if and when appropriate."
