The incdent happened around 12.30am on Sunday 5th August outside a convenience store in Salters Road.
Two men aged 34 and 36 were using the ATM outside the shop when they were attacked by three men.
Both victims have sustained various injuries with inquiries now ongoing to trace those responsible.
Anyone with information is asked to come forward.
The first suspect is white, of medium build and six foot three ins tall, with black-rimmed glasses and wearing black jeans and T-Shirt.
The second suspect is also white, between 5ft 7ins and 6ft tall with a skinny build, ginger spiked hair and wearing a grey track suit.
The third suspect is described only as wearing shorts.
Detective Constable Debbie Duncan from the Lothians and Scottish Borders CID said: “This was an unprovoked attack on a man using the cash machie, and when his friend went to his aid, he was also assaulted.
“Anyone who witnessed this assualt, or has any information that can help identify the suspects, should contact police immediately.”
Those with information can contact Lothians and Scottish Borders CID via 101 and quote incident number 604 of the 5th August. Alternatively, an anonymous report can be made to the charity Crimestoppers on 0800 555 111.