IAG boss Willie Walsh has been pursuing the Dublin-based carrier for months and had a third takeover bid, valuing Aer Lingus at €1.36bn (£1bn), accepted by the airline in January.
Today, though, agriculture minister Simon Coveney said IAG’s latest bid wasn’t persuasive enough.
"It is up to IAG to make an offer that is persuasive in relation to the public interest here, and as of yet we don't have an offer that is persuasive enough," he said.
Ireland’s government is likely to examine how a takeover would affect Aer Lingus workers, its 79-year-old brand and competition on routes to and from Ireland.
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.
The parent company of British Airways, which also owns Spain’s Iberia, has said it intends to operate Aer Lingus as a separate business with its own brand and management.
Aer Lingus is due to release its financial results for 2014 next Tuesday.
Shares in IAG fell 1.6% to 555p.