Union leaders have threatened to hold a strike ballot at the BBC if it presses ahead with its plan for 2,050 redundancies at the corporation.
Mr Thompson has announced 3,780 job cuts in all
The National Union of Journalists, Bectu and Amicus called on the BBC's managers to agree to three demands at a joint meeting on Thursday.
The unions demanded a three-month moratorium on the plans, which are aimed at saving £355 million a year.
If the BBC does not agree by 4 April, the unions will ballot their members.
The result of a ballot could be made known shortly before the general election, widely rumoured to be on 5 May.
The redundancies were announced by director general Mark Thompson on Monday.
Overall the BBC is shedding 3,780 jobs, 19% of its workforce.
Union bosses have demanded managers stop asking for volunteers for redundancy, and enter into negotiations so the proposals can be properly evaluated.
It is also seeking a guarantee there will be no compulsory redundancies and that pay and conditions will be protected for those whose departments are to be outsourced.
"There is a strong feeling of anger among staff over these job losses," said Gerry Morrissey, assistant general secretary of Bectu.
"They believe the director general has lost touch with what should happen at the BBC.
"A lot of these decisions appear to be politically motivated and the director general, chairman and BBC governors have decided to beat themselves up before the government does so."
Unions have warned the BBC could face "severe" industrial unrest over the massive job cuts announced last December and earlier this week.
Jeremy Dear of the National Union of Journalists said: "The BBC management have a window of opportunity to enter meaningful talks with staff representatives. If they slam the window shut, we will act.
"This is a grotesque display of self-harm being inflicted across the whole of the BBC at the hands of an incompetent management."
The latest cuts will be made across the production divisions, from TV and radio to news and new media.
The BBC was not available for comment.