Aer Lingus has said that it would consider engaging with the IMPACT trade union, if the company's pilots withdraw their threat to strike next Tuesday and Wednesday.
However, IMPACT Assistant General Secretary Michael Landers has described the Aer Lingus statement on possible discussions as 'insincere'.
He said the union was not ruling out the possibility of further work stoppages at Aer Lingus in the coming weeks.
The pilots claim the company wants to employ pilots for its new base in Belfast on less favourable terms and conditions than those in Dublin.
Aer Lingus has also said it has managed to lease aircraft and crews to operate a service for around 8,000 of its customers next Tuesday and Wednesday.
The limited services will operate between Dublin and Heathrow, Dublin and Manchester, Dublin and Faro, Dublin and Malaga, and Dublin and Amsterdam.
Limited services will also operate between Cork and Heathrow and Shannon and Heathrow.
Aer Lingus' commercial director Enda Corneille said the company had managed to lease aircraft which would accommodate around 8,000 of the up to 45,000 passengers who hope to travel with them on the days in question.
He said the company still hoped to source a limited transatlantic service and would have more details in the next 24 hours.
Earlier, the chief executive of the Labour Relations Commission, Kieran Mulvey, said that both sides in the Aer Lingus dispute have backed themselves into corners.
Speaking on RTE's Morning Ireland, he said they needed to be around the table in a framework for settlement. Mr Mulvey added that he feared that the future of Aer Lingus was at stake now.
Mr Mulvey said the Labour Court last June did not have the specific issues of Belfast or Shannon before it, so this was in effect a new dispute, and the services of the commission were available to Aer Lingus and to IMPACT and the pilots' union.
He said he believed there was time for the parties to sit down and come to some interim arrangement - if necessary - without putting at risk any commercial decision the company has made about Belfast.
He said the kind of 'Monty Pythonesque slapstick' engaged in by Ryanair chief executive Michael O'Leary could not be taken seriously, saying Mr O'Leary had a different business agenda which is not in Aer Lingus's interest.
