The university said donations made a "significant impact" to the institution.
It received $16.7million from alumni and friends last year.
However, it has dismissed calls for an external investigation, saying its present financial situation is not the result of mismanagement.
Alumni of the University of Otago in America (AUOA) former president Neil Matheson said alumni had discussed their commitment to the university and what they could do to make themselves heard.
"Based on the financial mismanagement, we are very reluctant to send any more money back to New Zealand to fund the university.
"Quite a few of us have decided to take the university out of our wills, and that means the financial support of very successful and, therefore, very wealthy alumni has now disappeared."
Mr Matheson, who was president of the organisation from 2016 until 2019, said AUOA raised more than $5million over a decade to fund programmes, such as study-abroad experiences in the US.
Alumni lives had been shaped by the great experience they had at the university, and they wanted to support students to go on and do great things, he said.
However, the university seemed unwilling to address their concerns.
"It’s really sad to see that an institution we love and respect so much is just going down the gurgler.
"It’s really a tragic situation."
Mr Matheson pointed to the university’s decision to spend $670,000 on its proposed new logo change as an example of poor fiscal management and "woke ideology".
A decision on the proposal is expected to be made at the university council meeting next month.
The university had an international reputation and it was critical to attract international students to boost revenue, Mr Matheson said.
"You don’t sell a product if you change the brand of it all of a sudden. People don’t know what the product is anymore."
It was a cop-out for the university — now facing up to several hundred staff cuts as it attempts to cut spending by $60million — to blame the situation entirely on Government underfunding and Covid-19, he said.
"They definitely need to bring in some outside help. They definitely need to get to the bottom of it."
Retired geologist and Canadian mining executive Clynt Nauman, who had contributed "a substantial amount" to the university, said major donations from alumni might diminish.
"Major donors are upset with the management for getting into this mess, with no contact, assurances or clarity given to us about how they are going to get out of it."
They wanted to support education but felt significant angst and uncertainty about the institution.
"There should be a decent forensic investigation by a leading accountancy firm to identify what went wrong and appropriate next steps."
He had contacted the university and been offered a meeting with a pro-vice chancellor, he said.
University chancellor Stephen Higgs said the Covid-19 pandemic had caused the biggest disruption to tertiary education since World War 2.
Government funding not keeping pace with inflation, young people travelling again, a buoyant job market and a cost of living crisis were also factors behind the current situation, he said.
Other universities were also dealing with these challenges, and mismanagement was not a factor.
"[The] university council has considered our current financial situation and has no concerns about the university’s financial management," Mr Higgs said.
Director of alumni relations Shelagh Murray said philanthropic donations from alumni and friends were an important part of the funding model.
"We are very grateful for the support we receive from our alumni as their contributions make a significant impact on the overall success and growth of our university," she said.
Alumni were provided with regular communications, including information about strategic direction Vision 2040 and its branding project, Tuakiritaka.
They had been invited to participate in a survey about the proposed change, and meetings had been held with many leading alumni and supporters.
"We are aware that some alumni do not support Tuakiritaka, but equally there are many who do." — Additional reporting Mary Williams
