Six cases caused by a bacteria similar to Mycobacterium lepromatosis have been confirmed in the squirrels since 2006.
It is the first time the disease has been found in the species and little is known about the spread of the potentially fatal disease.
Conservationists, however, say there are other more significant threats to red squirrels such as squirrelpox.
Scientists are urging the public to report sightings of squirrels which may be suffering from leprosy.
They said the information would help them build up a more detailed understanding of the disease.
There is no suggestion of any risk to human health.
Infected animals have been found from Dumfries and Galloway to the Moray Firth.
Symptoms include hair loss and severe swelling to the snout, eyelids, ears and feet.
Prof Anna Meredith of the Royal Dick School of Veterinary Studies at Edinburgh University is leading the research.
She told BBC Scotland: "We suspect this disease is more widespread than the six cases we have confirmed.
"Red squirrels are in decline. They are threatened by the grey squirrel and already face the major threat of the squirrelpox virus.
"This is the last thing that they need - another disease which could potentially threaten the population."
However conservationists from Saving Scotland's Red Squirrels said the leprosy infection appeared to be extremely rare and would not be a major threat to the wider population.
Project manager for Mel Tonkin, said : "Much more of a threat is the squirrelpox virus which is virtually always fatal to red squirrels, but finds a reservoir in grey squirrels.
"Research indicates that squirrelpox disease is responsible for accelerating the rate with which grey squirrels have replaced red squirrels throughout most of England and Wales.
"However, the conservation community has shown that red squirrel populations can be protected in the midst of grey squirrels carrying the virus by keeping the grey squirrel population density at a very low level."
Researchers in Edinburgh investigating the leprosy infection said dead red squirrels could be sent to them by post provided they were appropriately packaged.
But they advised members of the public to follow basic hygiene rules before and after handling dead squirrels.
Details can be found on this webpage under the "post-mortem guidance note section".
Scotland's red squirrels are facing a new threat from a form of leprosy, according to scientists in Edinburgh.