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It comes as thousands more cabin crew and ground staff are expected on Friday to receive letters from the airline telling them whether or not they are being made redundant.
Many of those who keep their jobs face pay cuts.
BA’s owner IAG announced in April that it would cut up to 12,000 jobs out of a total of 42,000 due to the effects of coronavirus.
In a statement, BA said coronavirus had had an “enormous financial impact” on the business.
“We are having to make difficult decisions and take every possible action now to protect as many jobs as possible,” the airline said in a statement.
“And, while we never could have anticipated being in a position of making redundancies, more than 6,000 of our colleagues have now indicated that they wish to take voluntary redundancy from BA.”
Five months after the coronavirus pandemic began, BA is still only running 20% of its usual flights.
Unite the union has launched a scathing attack on the BA, accusing the company of “industrial thuggery” and of forcing out loyal staff because of “company greed”.
Howard Beckett, Unite’s assistant general secretary, said: “This is a very bleak day for the incredible BA workforce and will go down in the history of the airline as the day that it put the interests of the boardroom ahead of its passengers and workforce.”