The patient, who is a bat conservationist, was admitted to hospital with symptoms of the potentially fatal infection.
A spokesman for NHS Tayside said he is in a "critical but stable" condition.
Rabies is a serious infection of the nervous system that is caused by a virus which is usually transmitted by a bite from an infected animal.
If confirmed as rabies, the case could be the first occurence of the infection acquired in the UK since 1902.
The health spokesman said: "The individual, who lives in Angus, was admitted with mild neurological symptoms which have since developed into a progressive neurological disorder consistent with rabies. Laboratory tests are under way.
"Clinicians at Ninewells believe the patient may have been infected by the European Bat Lyssavirus (EBL) - a type of rabies known to exist in several north European countries.
"The patient at Ninewells Hospital is known to have had prolonged close contact with bats over many years and had been bitten on at least one occasion."
Doctors said diagnosis is very difficult because the particular type of rabies being investigated is so rare.
Drew Walker, NHS Tayside's public health director, said: "It is important to stress that this is not a confirmed case of rabies.
"In the circumstances, however, we believe that it is prudent and sensible to regard it as such and take all necessary public health measures."
"One of the most important measures is to emphasise that the only members of the general public who might be at risk from EBL are those who handle bats or who have been scratched or bitten by them.
"Nobody else is at risk."
Professor Bill Reilly, a vet surgeon working for the Scottish Centre for Infectious and Environmental Health (Scieh), has been offering advice and support to NHS Tayside.
He said: "It is important to stress that this is not the traditional rabies virus as we know it.
"The unconfirmed case in Dundee concerns a rabies-like virus known only in bats.
"It doesn't pose a risk to the general public. Those who regularly handle bats in the UK are more at risk, but since 1996 such people have been advised to be vaccinated against this virus."
Prof Reilly said he was not aware of any cases where people had survived after being infected with bat rabies.
Conservation body Scottish Natural Heritage has advised all of its workers to avoid contact with bats until advised otherwise.
Disease specialists
Last month, the Department for the Environment, Food and Rural Affairs (Defra) warned that bats in the UK may be carrying the strain of rabies which can affect humans.
Tests carried out on a bat from Lancashire revealed the presence of the rabies strain European Bat Lyssavirus 2 (EBL 2) which in rare cases can be harmful to humans.
The Daubenton's bat was sent to the Veterinary Laboratories Agency in Weybridge, Surrey, to be analysed after it bit a woman on the hand in Lancashire on 11 September.
The woman, a bat conservationist, was examined by infectious disease specialists but did not appear to be suffering from any rabies symptoms.
An NHS Helpline - 0800 783 5066 - has been set up in Tayside to offer reassurance and advice.