Officers discovered the ammunition in the Blackford area of the Capital last night, and the items have been sent for further analysis.
The “significant” finding comes a week on from three incidents in Gilmerton and The Inch, where a Range Rover belonging to Dionne Hendry, the wife of killer Jamie Bain, was shot at and a homemade bomb wrapped in a Liverpool FC scarf was placed on the doorstep her home in The Murrays Brae.
Another device was left outside a house in Walter Scott Avenue, and at around the same time shots were fired at a property in Gilmerton Dykes Drive.
The Organised Crime Unit has also linked the incidents to a drive-by shooting in Fernieside Avenue earlier this month.
Detective Chief Inspector Kenny Gray said: “The recovery of the firearms ammunition on Saturday evening is significant, as it sends a clear message to those involved in serious organised crime that we are continuing with our efforts to shut down their illegal activities.
“Incidents like those witnessed last week remain extremely rare, but can cause significant distress to the local communities when they do occur.
“Police Scotland is committed to bringing those responsible for firearms offences and the intimidation of the public to justice.
“Whenever we receive intelligence relating to serious organised crime we will respond accordingly with all the necessary resources.
“If you have any information about the recent firearms incidents, or any other criminal activity then please contact police immediately.”
Officers were carrying out high-visibility patrols at the crime scenes in The Murrays Brae, Gilmerton Dykes Drive and Walter Scott Avenue last night to reassure the public.
Three men aged 18, 19 and 22 were arrested and charged following a series of early morning police raids on Friday.
They will appear at Edinburgh Sheriff Court tomorrow while a 52-year-old man will appear in court at a later date.
Anyone with information should contact Police Scotland on 101 or call the charity Crimestoppers anonymously on 0800 555 111.