The man threatened a female member of staff at the Bank of Scotland on Dumbarton Road, Scotstoun, during the attempted robbery, shortly after 3.30pm on Monday.
Police described the suspect, who left the bank empty handed, as a white male in his late 30s or 40s wearing a black shoulder length wig, white baseball cap, tweed suit jacket, red tartan scarf and blue jeans.
He had a Scottish accent and was of slim build with a gaunt appearance.
The suspect appeared to be in possession of a firearm, police said.
He ran away in the direction of Harland Street and was last seen in the area at Gleneagles Lane.
Police believe he may have been in the same area, near to the bank, before he entered the premises.
Officers have appealed for witnesses and are examining CCTV to identify the person.
Detective Constable Alister Thom, from the Robbery Unit at Pollok, said: “We are examining CCTV in an effort to identify the person responsible.
“It is believed that he may have been in the area of Harland Street near to the Bank before he entered the premises.
“He also headed off in that general direction and was last seen in Harland street at Gleneagles Lane.
“Dumbarton Road is always busy with traffic and pedestrians and I am sure that people would have seen him in the area.
“His clothing, together with the wig, would have made him quite distinctive and I would urge anyone who saw him to come forward.”
Anyone with information should contact Pollok Robbery Unit on the non-emergency number 101 or call Crimestoppers on 0800 555 111 where anonymity can be maintained.