IAG, which also owns Spanish airline Iberia, has pitched its latest offer at €2.55 per share – valuing the airline at €1.34bn (£1bn).
The improved bid follows the rejection of two earlier offers of €2.30 and €2.40 a share.
Success of the latest approach is likely to depend on whether Willie Walsh, IAG’s chief executive, can persuade the Irish government to support the deal.
The Irish state owns 25.1% and Paschal Donohoe, the transport minister, is expected to brief the cabinet on any takeover proposal on Tuesday.
He has reportedly warned that the value of the government stake — about €325mln based on IAG’s latest offer — was not the only issue that would be considered.
“I and the government will take huge care in evaluating any proposal that comes in,” he told state broadcaster RTE on Sunday.
IAG is said to be eyeing Aer Lingus’s slots at Heathrow as the Dublin-based carrier is the third biggest airline at the airport.
It is thought that the deal may be blocked by the government if it does not include guarantees that Aer Lingus will continue to use its slots at London’s Heathrow airport to fly to and from Ireland.