At least six of the creatures have died after catching the disease, which has never been seen in UK wildlife before.
Researchers at Edinburgh University said the disease was caused by microbacterium lepromatosis, which does not affect greys.
Cases reported so far stretch from Dumfries and Galloway to the Moray Firth.
Infected red squirrels suffer swelling and baldness around the nose, ears and feet, and “look really quite bizarre”.
Professor Anna Meredith of the Royal Dick School of Veterinary Studies at the university confirmed there were six recorded cases so far.
She said: “It is definitely a form of leprosy which we’ve never had described in wildlife in the UK.
“We really don’t know anything about the disease yet.
“These cases are scattered across Scotland, there doesn’t seem to be any pattern.”
She added: “There may be many other affected squirrels out in more remote areas that we’re just not picking up.”
Appealing to the public for help, Prof Meredith said anyone finding a dead red squirrel should post it to the Dick Vet, so long as it was appropriately packaged.
The same bacteria causes leprosy in humans, although at present there is no evidence that it will spread to any other species.
Researchers admit they do not yet know if the leprosy itself is actually killing the squirrels.