Armed officers were sent to Chester Street in the city's west end at about 20:00 on Wednesday.
Welsh was made famous after playing a gangland boss in Trainspotting 2 who owns several seedy saunas around Edinburgh.
Edinburgh Evening News editor Liam Rudden honored Welsh on Twitter writing, "Remembering Bradley Welsh as a man who was passionate about what he believed in and his community".
A police spokesman said: "Police in Edinburgh received a report of a disturbance in Chester Street at around 8pm on Wednesday 17th April". Officers will continue to provide a high-visibility presence in the area to engage with the public, offer reassurance and gather information that can assist with this investigation.
He previously played for Hibernian as a casual before turning his hand to boxing and is a past British Amateur Boxing Association champion.
The star had also appeared in Danny Dyer's Deadliest Men back in 2008, and The Real Football Factories back in 2006.
"He was one of a kind and will be sadly missed".
Tributes were paid on social media, with one woman posting a picture of her and friends alongside Welsh in a boxing ring.
"I told Danny he could film in my gym which he said would be 'perfect". Around three police cars and a black van drove along the street and the traffic then stopped. "Before doing so we heard the police say to the National incident response unit - gunshot would (sic) to the head".
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