British Airways says it aims to operate a full long-haul schedule and a “high proportion” of short-haul services after the outage caused by a power failure.
It says passengers should check the status of flights before travelling.
Cancellations and delays affected thousands of passengers at both Heathrow and Gatwick on Saturday.
All flights operated from Gatwick on Sunday but more than a third of services from Heathrow – mostly to short-haul destinations – were cancelled.
In a statement, BA said its IT systems were moving “closer to full operational capacity”.
“We continue to make good progress in rebuilding our operation, following Saturday’s major IT systems failure which severely affected our operations worldwide,” it added.
“At Heathrow, we have operated virtually all our scheduled long-haul flights, though the knock-on effects of Saturday’s disruption resulted in a reduced short-haul programme.
“We apologise again to customers for the frustration and inconvenience they are experiencing and thank them for their continued patience.”
British Airways is liable to reimburse thousands of passengers for refreshments and hotel expenses, and travel industry commentators have suggested the cost to the company – part of Europe’s largest airline group IAG – could run in to tens of millions of pounds.
Customers displaced by flight cancellations can claim up to £200 a day for a room (based on two people sharing), £50 for transport between the hotel and airport, and £25 a day per adult for meals and refreshments.
On Saturday night, travellers spent the night sleeping on terminal floors at Heathrow on yoga mats provided by British Airways.
The disruption continued into Sunday, with queues building up as passengers tried to rebook flights. Conference rooms at the airport were opened to provide somewhere more comfortable for passengers to rest.
British Airways said Heathrow was still expected to be congested on Monday and urged travellers not to go to the airport unless they had a confirmed booking for a flight that was operating.
It said passengers could get a full refund or rebook to travel up to the end of November but recommend they use its website.
The airline said there was no evidence the computer failure was the result of a cyber attack. It denied claims by the GMB union that problem could be linked to the company outsourcing its IT work.
Gatwick Airport said it was continuing to advise customers travelling with British Airways to check the status of their flight with the airline before travelling to the airport.