BA said it aimed to operate a near normal schedule of flights from Gatwick airport and the majority of flights from Heathrow on Sunday.
However, Heathrow Airport said it expected further delays and cancellations of British Airways flights on Sunday and told passengers not to travel to the airport unless they were rebooked on other flights.
"We are continuing to work hard to restore all of our IT systems and are aiming to operate a near normal schedule at Gatwick and the majority of services from Heathrow on Sunday," BA said in a statement.
"We are extremely sorry for the huge disruption caused to customers."
British Airways cancelled all its flights from London's Heathrow and Gatwick airports on Saturday after a power supply problem disrupted its flight operations worldwide and also hit its call centres and website.
Alex Cruz, the chairman and chief executive of BA, part of Europe's largest airline group IAG, said there was no evidence of any cyber attack. He said a power supply issue was to blame.
Thousands of passengers were left queuing for hours in departure halls at the airports on a particularly busy weekend. There is a public holiday on Monday and many children were starting their school half-term breaks.
Terminals at Heathrow and Gatwick became jammed with angry passengers, with confused BA staff unable to help as they had no access to their computers, according to passengers interviewed by Reuters.
"We are refunding or rebooking customers who suffered cancellations on to new services as quickly as possible," BA said, adding that it had introduced more flexible rebooking policies for passengers affected.
While other airlines have been hit by computer problems, the scale and length of BA's computer problems were unusual.
Delta Air Lines Inc cancelled hundreds of flights and delayed many others last August after an outage hits its computer systems.
Last month, Germany's Lufthansa and Air France suffered a global system outage which briefly prevented them from boarding passengers.
(Additional reporting by Kylie MacLellan; Editing by Susan Fenton)
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)