In a note to staff posted on the broadcaster’s internal network, seen by Yahoo News UK, director-general Lord Tony Hall said the BBC was already facing financial pressures due to the government’s decision to stop funding licence fees for people over 75.
Around 3.7 million pensioners who previously received a free TV licence will have to pay when new rules come into force in August.
But anyone who receives a pension credit will remain eligible for a free licence, costing the BBC around £250m a year.
Lord Hall said the coronavirus pandemic has caused the corporation additional financial pressures.
The BBC now has 24% less to spend on its UK public services than if the licence fee had risen with inflation over the last decade, Hall added.
The upshot, Hall said, is the corporation must save an extra £125m this year – and even more next year.
In a call to all staff, he said: “We know hard choices are necessary. Over a third of our costs – across the BBC – relate to our people.
“That’s why we’re introducing this voluntary redundancy programme.”
All UK-based public service employees are to be given the chance to express interest in taking voluntary redundancy.
A BBC spokesperson said: “The impact of the coronavirus pandemic means the BBC needs to make £125m of savings this financial year, in addition to the considerable efficiency savings the corporation had previously committed to and planned for.
“The BBC’s challenge is to keep delivering programmes and services for the whole country while continuing to adapt and change.
“The BBC is therefore inviting public service staff to express an interest in voluntary redundancy.
“This is a necessary process to ensure the BBC meets the challenges of a fast-changing media environment within its financial perimeters.”
In 2015 the BBC had to take over the cost of providing free licences for over-75s by 2020 as part of its latest funding round.
But it decided to start charging some over-75s because it said it could not afford to provide licences without the government subsidy.
The BBC delayed the implementation of the charge from June to August this year due to the coronavirus pandemic.
BBC chairman Sir David Clementi said: “The BBC board has decided to delay changes to over 75s’ licence fees. We are in exceptional circumstances.
“Now is not the right time. We are fully focused on delivering our services to the public at this difficult time.”
It was revealed on Monday the number of people on employers’ payrolls fell by more than 600,000 in April and May as the coronavirus lockdown hit the labour market.
Vacancies also plunged by the most on record, data showed on Tuesday.