British Airways (BA) said that it is aiming to run a "near normal schedule" at Gatwick and the "majority of services" from Heathrow on Sunday after a "major" IT failure resulted in the cancellation of all flights.
Serious problems with BA's systems led to thousands of passengers having their plans disrupted on 27 May, the BBC reported.
The airline said there was no evidence the computer problems were the result of a cyber attack.
The company's chief executive Alex Cruz had said it was believed "the root cause was a power supply issue".
The IT failure had affected check-in and operational systems, including customer service phone lines.
BA said although some of its IT systems have returned, "there will be some knock-on disruption to our schedules as aircraft and crews are out of position around the world.
"We are repositioning some aircraft during the night to enable us to operate as much of our schedule as possible throughout Sunday,"the BBC quoted the airlines as saying.
Passengers described "chaotic" scenes at the airports, with some criticising BA for a lack of information.
The airline apologised and said it was refunding and rebooking customers.
"We are extremely sorry for the huge disruption caused to customers throughout 27 May and understand how frustrating their experiences will have been," a BA spokesperson said.
"We are refunding or rebooking customers who suffered cancellations on to new services as quickly as possible and have also introduced more flexible rebooking policies for anyone due to travel on Sunday and Monday who no longer wishes to fly to/from Heathrow or Gatwick."
BA advised customers to continue checking the status of their flight on its website www.ba.com before travelling to the airport.