The state broadcaster began meeting with staff on Tuesday over its potential plans to cut costs.
At least 68 jobs were proposed to be cut when TVNZ announced the plans last month.
Staff from the youth-oriented programme Re News were due to have a meeting starting at midday.
Fair Go began screening in 1977.
TVNZ is to meet staff of current affairs programme Sunday tomorrow to confirm its future. An all-staff meeting is due to be held that afternoon.
"So it's a challenging day for staff at TVNZ coming up, but they and we remain determined that we'll continue fighting to save our stories so we continue to have good quality news and current affairs at TVNZ."
He did not want to speculate on what management would say, but staff had worked "really hard" to come up with some counter proposals.
No one was denying it was a challenging environment for TVNZ, however, staff had worked in teams to come up with some fresh ideas to avoid programmes being axed and jobs ending.
Wood said it wouldn't be helpful to make public the details of what staff had suggested.
"Broadly speaking, staff at all of these shows have worked hard to develop proposals to retain that news and current affairs capacity ... to find some ways that perhaps some of these shows perhaps could be done a bit differently."
Staff were willing to meet the challenges of helping TVNZ generate more revenue but wanted more say "and genuine engagement."
"Because to be honest just cutting news and current affairs now isn't going to solve TVNZ's underlying revenue challenges."
Many viewers were also supportive, he said.
"We've received huge support from thousands of New Zealanders for the 'Save Our Stories' campaign ....Kiwis want to keep these iconic shows ...we really hope TVNZ listens to that."
The union maintained TVNZ had not followed its legal obligations under the collective agreement with staff, and legal action was being considered if management did not have a change of mind over the job losses.
Newshub meeting on Wednesday
Staff at Newshub on channel Three are expecting to be told at 11am on Wednesday if the service will close in June with the possible loss of up to 300 jobs.
Warner Bros. Discovery proposed the closure in February as a cost-cutting measure and has been consulting with staff on alternatives to the closure.
It has been reported that other media companies are interested in acquiring Newshub and continuing a scaled-down news service, but no details of any negotiations have been confirmed.
