The university, initially funded entirely through charitable donations, was once viewed as a groundbreaking model for free tertiary education.
It started running into financial difficulties, and in 2012 was placed in business rescue and then in provisional liquidation last year.
According to court papers, it is almost R42m in the red.
The 56 staff members who have gone to court are among the university's creditors and are owed about R8m in unpaid salaries. They say there is an offer on the table that would keep the university going, and also put it on a sounder financial footing. Yet, inexplicably - they claim in court papers - the university's liquidators have declined the offer, on the basis that it was subject to too many conditions. One of the liquidators, Johannes Muller, said they would respond in court papers.
The staff members have asked the court to order the liquidators to urgently convene a meeting of creditors to "consider and vote on" any offers to buy the institution - in particular the offer by Africa Integras, a company specialising in developing African educational infrastructure.
On behalf of the staff, Daniel Mncube said the Africa Integras offer would allow the university to continue "to operate ... with the spirit and purpose for which it was created".
In an affidavit, Lord Joel Joffe - chairman of the Joffe Charitable Trust, another of the university's creditors - said the trust had applied only for provisional liquidation "in desperation", after hearing that teachers had gone without their salaries for three months.
The idea was to ensure the institution would be restructured to run on "sound financial principles for the benefit of its students, future generations of students, and its staff" - not to close it down, Lord Joffe said.
Mncube said: "We believe the liquidators are not acting in the best interest of the staff, the students or the public by refusing to... negotiate with Africa Integras", and that they had breached the constitutional right to education and the Companies Act.
Instead they had acted in "an increasingly uncompromising and obstructive manner", appearing to favour selling off Cida's "considerable" assets. Mncube said it was standard business practice for deals of this nature to be subject to suspensive conditions.
"The failure of experienced liquidators to comprehend this is difficult to understand."
He said if the university were to shut down, its creditors would have to bear the cost of finding alternative educational arrangements for its students.
Lord Joffe - one of Nelson Mandela's attorneys during the Rivonia trial - said Africa Integras's offer was a "convincing proposal". It included significant financial support from a group of prominent international and local financiers.
Lord Joffe said he was not sure why the liquidators had taken the actions they did. But he suspected there was "personal animus" against Africa Integras's MD Andrea Pizziconi and the company's team.
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