The current five-day strike by BA cabin crew has entered its fourth day after fresh talks failed to reach a solution.
The latest talks between the company and the Unite union, which represents cabin crew, broke up late on Tuesday without agreement.
There was no indication as to whether progress had been made.
A spokesman for the conciliation service, Acas, said it would be be in touch with both sides to arrange a future meeting date.
The current strike is the second of three five-day walkouts planned by BA cabin crew.
With another day to go in this block of strike action, BA has already said it is prepared for the next five-day run of industrial action, which is due to start on Saturday, and throughout that plans to fly more than three-quarters of customers who booked flights.
It has also said it expects to operate all flights to South Africa ahead of the World Cup.
These departures will be part of the 80% of long-haul flights it aims to operate from Heathrow.
It also expects to run 60% of short-haul services from the airport. Flights from Gatwick and London City airports will continue to be unaffected.
The disagreement between Unite's cabin crew workers and BA centres on jobs, pay and working conditions.
One key sticking point is the travel perks given to crew. Unite is demanding that BA reinstates the travel perks of striking workers, and cancels disciplinary proceedings against others.
The union has offered to call off the strikes if the travel concessions are put back in place, ahead of agreeing a final deal.
BA says it has offered to reinstate travel concessions, but only as part of a final agreement.
On Monday, Unite warned it could ballot members for further walkouts in July. BA has yet to comment on this threat.