The 85-year-old woman was left “shocked and distressed” after having her handbag snatched by two teenagers just off Leith Walk.
Officers have not ruled out the possibility that the incidents are linked and are appealing for witnesses to both of the muggings.
The elderly woman was walking alone along Brunswick Road on Friday at 5.30pm when the two boys – believed to be around 16 or 17 – stole her bag and then ran off towards Leith Walk.
It followed a similar incident last Tuesday when a 67-year-old woman was mugged and pushed to the ground.
The retired shop worker – who was left bruised and suffering from shock – had been on her way to visit a friend when the attack took place in nearby Brunswick Street at 5pm.
In the latest incident, officers recovered the elderly woman’s handbag from the railway line at Dicksonfield a short time after it was stolen, but the contents – which included an undisclosed sum of money and personal items – had been removed.
Police say they are particularly keen to speak to a woman who was in her green hatchback car on Allanfield, as they believe she may have seen the suspects snatch the bag before fleeing the scene.
The suspects are said to be 5ft 10ins tall with athletic builds.
Both were wearing baseball caps and one was wearing a dark-coloured tracksuit while the other wore a grey tracksuit top and red tracksuit bottoms.
A police spokesman said: “The woman was not assaulted during the theft and so was uninjured.
“However, she was very shocked and distressed at what happened.
“Her bag contained a sum of cash and items that were of personal or sentimental value to her and she is eager to have these returned.
“Anyone who was on Brunswick Street and witnessed this incident is asked to contact police immediately.
“Similarly, the woman in the green car or anyone else who can assist with our inquiries is also asked to come forward.”
The two muggers in the first incident on Tuesday – who escaped with the woman’s handbag containing £130, a bank card and a mobile phone – ran away towards Montgomery Street.
There is no description of the suspects in this incident as they escaped too quickly for their victim to catch sight of them.