Aer Lingus has rejected an approach from British Airways-owner International Consolidated Airlines Group.
The Irish airline said that IAG had made a "preliminary, highly conditional and non-binding approach" which it rejected two days ago.
Aer Lingus said its board felt the proposal "fundamentally undervalues" the airline.
IAG - which also runs Iberia and Vueling - confirmed it had made an informal takeover approach for Aer Lingus, adding that there was no certainty a further proposal or offer will be forthcoming.
"International Consolidated Airlines Group notes the recent movement in the share price of Aer Lingus and confirms it submitted a proposal to make an offer for the company, which has been rejected by the board of Aer Lingus," IAG said in a brief statement that gave no indication of price.
Shares in Aer Lingus had jumped in early trade on the back of a report in the Financial Times that IAG was considering a takeover bid.
Aer Lingus shares were up 14% by mid-afternoon while shares in IAG - which also runs Iberia and Vueling - were up by 3.3%, after the Financial Times story, which cited "people familiar with the situation".
Following statements from both groups, Aer Lingus' share price moderated somewhat, to close about 9% higher in Dublin trade.
Buying Aer Lingus would give IAG additional slots at Heathrow Airport, the main British Airways hub in London which is operating close to its capacity.
However any acquisition of Aer Lingus would need the backing of Ryanair - which owns a 29.9% stake in Aer Lingus after three failed takeover attempts - and the Government, which owns 25% of carrier.
Such an acquisition could also raise competition concerns, due to the overlap in the two airlines' service.
Aer Lingus Regional sees 20% growth in passengers
Aer Lingus Regional passenger numbers are 20% ahead of the same time last year.
The airline, which is operated by Stobart Air for Aer Lingus, carried just over 1.2 million passengers between January and the end of November.
November was the 17th consecutive month of growth for Aer Lingus Regional.
It flew 89,000 passengers during the month on routes including Dublin to Newcastle and Cork to Glasgow.
Passenger numbers to Glasgow were up 36% on November last year thanks to the Ireland v Scotland Euro 2016 qualifier during the month.
The airline said it was benefiting from demand for trans-Atlantic flights from passengers connecting through regional airports.
Aer Lingus regional's Dublin to Kerry service grew traffic figures by 22% year-on-year with many of the passengers either heading to North America through Dublin or coming from the US and Canada and connecting onwards to Kerry.