The Aer Lingus board is considering a revised €2.55 a share offer from IAG, the owner of British Airways.
The new €1.36bn offer, which follows two previous bids, would depend on the recommendation of Aer Lingus's board and both Ryanair and the Irish government.
Ryanair has a 29.8% stake in Aer Lingus, while the government owns 25.1%.
Aer Lingus had rejected two bids from IAG last year.
IAG's primary interest in the Irish carrier is its 23 pairs of take-off and landing slots at Heathrow airport - valued at about £30m apiece - allowing it to operate more flights.
Aer Lingus is the fourth-largest operator at Heathrow after BA, Lufthansa and Virgin Atlantic