The 52-year-old father-of-two was killed during a dawn raid at a block of flats near the city centre as part of an “intelligence-led operation” in the early hours of Friday morning.
Mr Alton Smith who is said to have lived alone, suffered a cardiac arrest and died at the scene.
One officer fired one shot at the flat and no one else was present, according to the Independent Office for Police Conduct (IOPC) which is investigating the incident.
The watchdog added that a “non-police issue firearm” was recovered from the property.
Friends paid tribute to the Birmingham City FC fan on social media, with some describing him as “Big T”.
Some appeared to reference the football hooligan firm Birmingham Zulus, although it is unclear whether Mr Smith had any connection with the group.
One Twitter post read: “I have just had a call from a ex Zulu that 1 of there lads back in the day got shot dead this morning by the old bill, (that’s all I no) Trevor Smith rip my friend from all at Chelsea [sic].”
While others read: “Awful news, one of Birmingham’s Finest Trevor Smith Big T RIP mate”, and “Trevor Smith, RIP my friend”.
IOPC commissioner Derrick Campbell said video footage from the body-worn cameras of officers at the scene had captured the incident, including the gunshot, and was being reviewed.
He added that the area was being forensically examined as part of the inquiry and that the officers involved had been taken to a “neutral police station” for debriefing.
“There were, within close proximity, in the region of six police officers,” he said. "We know from early indications there were two officers standing in front of the door.”
West Midlands Police said in a statement: “Enquiries are under way after a man was shot dead by armed police during an operation at an address in Birmingham. A man, in his 50s, was shot and died a short time later at the scene.”
It is the second fatal shooting involving the force’s armed officers this year, after 31-year-old Sean Fitzgerald was shot once in the chest during an incident in Coventry in January.
Commenting on the recent shootings involving police, Mr Campbell said: “Fortunately within the Midlands in the last 17 years we’ve had two fatalities – unfortunately [it is] in the last three months those fatalities took place.”
He added: “The public want to be assured the police officers carry out their job without impunity and in a professional manner and that is the reason we are involved to ensure a robust investigation is taking place.”
An ambulance spokesperson said CPR was being performed on Mr Smith when crews arrived at the flat on Friday.
Ambulance staff took over but despite their best efforts he was confirmed dead at the scene.