The black and white image, which has appeared on a wall in Grindlay Court, just off Bread Street, seems to show a young girl looking down at a mobile phone.
Kate Hermiston, 24, spotted the mural last night and again this morning as she made her way to work in the nearby Medusa hair salon .
She told the Evening News:: "It definitely wasn't there yesterday morning but was there last night and again this morning.
"I don't know if it is a Banksy or not, but I know there was a lot of talk about Massive Attack were playing in Glasgow on Monday night, and there have been rumours that Banksy is associated with them.
"I saw it for the first time last night. A lady came into the salon and asked if we'd seen it because it's only over the road.
"We all walked over to take a look and we think it could be a genuine Banksy.
"It definitely looks like a Banksy, but we don't know if it is just a copycat."
The senior stylist added: "I'm a fan of art and I think it's great. Banksy is a very good graffiti artist and I'm just hoping it is an original.
"It will encourage people to come to this area of the city. It is in a discreet location and it is the first Banksy I have seen in Edinburgh. It could be the first one ever in Scotland."
If confirmed, the image could be highly valuable. Last week the owner of a garage in Wales sold a Banksy mural painted on his wall for over £100,000.
Ian Lewis, 55, turned down larger offers to sell the piece to an art collector who promised to keep it in Port Talbot, South Wales.Street artist Banksy confirmed he was behind the image, which appeared before Christmas.
A mural attributed to Banksy was stolen last weekend from the Bataclan concert hall in Paris. The painting, of a ghostly veiled figure with a downcast gaze, is believed to be a memorial to the 90 people who died there in a 2015 terrorist attack.
According to Jeux, the robbery took roughly 10 minutes. "The painting was even protected by a Plexiglass. The whole door disappeared," Jeux said. "The security alarms went off and our security agency immediately alerted us."