Members of broadcast union Bectu and the National Union of Journalists have asked BBC staff to show their opposition for cuts and privatisation on March 2 - the day the government's long-awaited Green paper on the future of the BBC is expected to be released.
So far up to 20,000 badges displaying the slogan "Fight For Our BBC" have been issued to members across the country.
Demonstrations and meetings across BBC premises, including the main studio at London Television Centre, have been organised.
Unions are warning of industrial action if compulsory redundancies take place under the cuts.
Thompson confirmed last December that 2,900 jobs are to go at the broadcaster, and 1,800 staff are to be moved north to Manchester as part of a far-reaching restructure of the corporation. It is believed that up to 6,000 jobs could be axed in total.
The cull was due to four reviews into internal operations at the BBC called shortly after Thompson took the helm in 2004.
The Green Paper, tipped for release tomorrow, will confirm the BBC has won its licence fee for a further 10 years.
Media secretary Tessa Jowell is said to be impressed with Thompson's cost cutting target of £320m.
However, she is not in favour of carrying out former BBC director general Lord Birt's "top-slicing" proposal of giving a certain amount of the BBC's funding to other commercial broadcasters.
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