The package, believed to have been offered to every member of the trust's staff, is part of a plan to cut 700 jobs over the next three years to save£40m.
The trust runs Darlington Memorial Hospital, Bishop Auckland General Hospital, the University Hospital of North Durham, Chester-le-Street Hospital and Shotley Bridge Community Hospital.
At Bishop Auckland, built five years ago for£67m, a Save Our Hospital group has already been set up following several ward closures.
Trade union Unison has urged staff to reject the deal, described by regional officer Ian Daley as a 'cheeky scheme' that undercut a national redundancy package negotiated with the NHS.
'We will be tell our members they should not have anything to do with this, scheme' he said. 'It will be used to get rid of people on the cheap.'
Mr Daley said the trust had been made aware of the union's opposition to the scheme, but had decided to press ahead.
A trust spokesman said the scheme was the a result of a drop in demand for inpatient care.
'Patients want to be treated closer to their homes. You can't argue with that, but we have to plan our response. We don't have any proposals for compulsory redundancies.'