Up to a dozen armed officers were sent to Chester Street about 8pm where they found a seriously injured man in the street.
The man, who died at the scene, is understood to be 42-year-old Mr Welsh.
The former Hibs casual turned charity worker played gangland kingpin Doyle in the film and was friends with author Irvine Welsh.
A Police Scotland statement said: “On arrival of emergency services, a man was found seriously injured in the street and sadly passed away at the scene.
“His death is being treated as suspicious and inquiries are continuing.
“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.
“Anyone who believes they have any relevant information should contact police immediately.”
Mr Welsh is a past British ABA Lightweight boxing champion.
He has run charity projects in the city to help young people stay fit and out of trouble through his Holyrood Boxing Gym.
Mr Welsh is also involved with Edinburgh charity Helping Hands.
In 2017, Mr Welsh secured a role in Trainspotting T2 thanks to a successful Guinness World Record attempt that saw him spar with 360 people in succession, including director Danny Boyle.
Two people are understood to have identified themselves as family members to police at the scene and demanded information.
Chester Street is in lockdown, with part of the road cordoned off from Walker Street to Manor Place.
Two fire engines arrived at the scene about midnight.
Multiple police vehicles are in attendance, including a dog unit.
Three ambulances and paramedics are also at the scene.
Officers have told residents to stay indoors, with the cordon having since been extended to Drumsheugh Gardens.
Resident Alasdair Morton, 46, said: “I came out the house and we were told to go back in. Around three police cars and a black van drove along the street and the traffic then stopped.
“I initially thought it was a police escort then when I had a look there must have been a dozen or so police with guns pushing the traffic back.
“We’ve not been told anything, but police waved through some ambulances.
“They said ‘there’s a gunshot wound to the head somewhere’. We could still hear noises that suggested there was a situation still going on.”