Both parties would be contacted to arrange more talks in an attempt to avert a further 10 days of industrial action planned during the coming weeks, it added.
Cabin crew are in the final day of a five-day strike, protesting over reduced staffing levels and cuts to benefits.
Seven days of walkouts in March cost BA 43 million pounds ($62 million).
The stoppages come at a difficult time for BA, which last week reported a second straight year of record losses and is battling a global economic downturn and industry-wide recession.
Ongoing industrial action, coupled with further disruption to flights in April caused by ash from an Icelandic volcano, could scupper BA's hopes of avoiding a third year of losses.
The loss-making carrier said that in the event of another 5-day strike next week, its longhaul schedule at London's Heathrow airport would be increased to more than 70 percent of flights, from 60 percent this week.
It aims to increase the shorthaul schedule at Heathrow to 55 percent of flights from 50 percent and operate a full schedule from London's Gatwick and City airports.
BA, which flies around 90,000 passengers a day, said about a quarter of its passengers would be affected by the strikes, but that they could claim a full refund, rebook or reroute their journey.
Previous negotiations in the long-running dispute have been acrimonious.
BA Chief Executive Willie Walsh and leaders of Unite blame each other for a breakdown in communication.
The issue of travel allowances for cabin crew has become a major sticking point in the conflict. Unite had offered to postpone the strikes if travel allowances for cabin crew were reinstated.
(Reporting by Rhys Jones and Avril Ormsby; editing by Paul Hoskins and Jon Loades-Carter)