The Learning and Skills Council is planning to make 1,120 staff redundant by June this year, and unions say it is using assessment centres and job-related tests to decide which employees they should keep.
The Public and Commercial Services union, which represents 1,700 of the LSC's 4,000 workers, voted to strike, calling the selection process "crude and brutal".
Mark Serwotka, general secretary of the union, said: "It is unacceptable that staff are being treated like children in being forced to sit 11-plus-style exams to keep their job."
Nearly 88 per cent of union members voted to strike, on a turnout of about 64 per cent.
A date for the one-day action has to be set within the next four weeks.
"This result illustrates how out of touch senior management are and the depth of anger they have provoked among staff by their scandalous approach to axing staff," Mr Serwotka said.
He said it was ironic that the LSC, responsible for funding pound;9 billion of training, was not trying to reskill and retrain staff rather than just forcing them to reapply for their jobs.
The union argues that the cuts are too extensive and too haphazard, and will mean the LSC will not be able to administer its budget effectively.
About 90 per cent of staff in the regions have already been formally told their post "ceases" in June, according to the PCS.
To end the strike threat, the union will demand that all redundancies be voluntary, that staff leaving get fair compensation, and that flexible working is introduced so remaining employees can maintain a work-life balance.
Rob Wye, the LSC's director of communications, said the strike had the support of just 30 per cent of the council's total staff.
The selection process was fair, transparent and open, he said, and the tests were designed to assess managerial ability.