Minority Democrats and some Republicans said they were still working to get the votes to override the 41% cuts by the Republican at a special sitting of the state legislature.
The University of Alaska has said the reductions – including $130 million from its own budgets – would force it to shut down programs and lay off up to 40 percent of staff.
Dunleavy, in his first year as governor, has said his budget vetoes are needed to scale back what he sees as bloated spending, and to cope with a long-term fall in state oil revenues.
Lawmakers needed 45 votes – three quarters of the state legislature – to overturn the cuts. Democrats, who have 22 seats, did not say how close they were to that target.
University of Alaska President Jim Johnsen wrote to students and staff on Friday, warning that the cuts would “strike an institutional and reputational blow from which we may likely never recover”.
Dunleavy, a former teacher, said the same day he believed the university was resilient and would emerge in a healthier state.
“I believe that they can turn the university into a smaller, leaner but still a very, very positive, productive university here in the northern hemisphere,” he said.
He has pushed through a total of $440 million cuts to state spending to pay for one of his chief campaign promises – an increase in the annual oil revenue dividend the state pays to Alaska residents.
Other cuts have targeted the Medicaid program, social services, law enforcement and services for the poor and elderly.
The International Arctic Research Center is based in the university’s Fairbanks campus.
By Yereth Rosen in Anchorage, additional writing by Steve Gorman and Rich McKay; Editing by Andrew Heavens