The man was found at around 5am on Sunday in Oxgangs Street and was taken to the Royal Infirmary.
He remains in hospital and police are seeking to determine how the man, who was only identified by officers yesterday, was hurt.
A police spokesman said: “At this time we do not know how he has come by his injuries and are keen to hear from members of the public who can help us fill in the blanks.
“Anyone who remembers seeing this man or has any information that can help us establish the circumstances that led to him being found injured is asked to contact police.”
The man is described as white, late 50s or early 60s, between 5ft 8in and 5ft 10in tall with a chubby build and a bald head with hair at the sides. He was found wearing black leather shoes, black cotton jogging bottoms and a navy woollen jumper. He also had a walking stick with him.
Anyone with information is asked to contact police on 0131-311 3131, or 0800 555 111.