The incident is said to have happened shortly after 5pm on Monday, with witnesses describing how a young man entered Kaspas Desserts shop on Gorgie Road covered in blood and asking for help.
Eyewitness Shaun Pinkerton said he and some staff from the shop gave the man first aid until the ambulance arrived, attempting to bind up the injured man’s leg.
Shaun, 28, said: “I was waiting on my ice cream when this guy who was about 18 came in, he was shouting for help.
"He appeared to have multiple stab wounds on his left leg and it was gushing blood everywhere. It was just horrible.
"I put him on a couch and shouted to staff to get me something to tie his leg with to keep the pressure on it.
"I asked him if he’d been stabbed anywhere else, but he didn’t know.
"The staff were helpful, and the ambulance was pretty quick.
"Police are still there, they were searching for something on the other side of the road.”
Another witness, one of two staff members who helped the injured man, said: “He came in and said he needed help.
"He said he’d been stabbed and needed an ambulance.
“I called the ambulance and got another colleague to help.
"We were just trying to do what we could until the ambulance got there.”
Police Scotland and the Scottish Ambulance Service have been contacted for comment.