Police sources have refused to deny Sunday newspaper reports that a man matching the description of the 48-year-old’s killer had been spotted near the scene of last Wednesday evening’s shooting.
A neighbour claimed he saw the man described by detectives in the former boxer’s West End street two weeks ago.
Welsh - who had close associates in the Capital’s underworld - was gunned down outside his £500,000 basement flat in Chester Street at 8pm on Wednesday.
The neighbour said he saw a man matching the Police Scotland description of the killer lurking nearby.
He told a Sunday newspaper: “He was a young man standing there on his own looking up and down the street very intently as if he was checking something out.
“When he saw me looking at him, he just walked off. I did not think anything of it at the time but I wonder now if he was casing Bradley’s house.”
Police said the suspect is aged 20 to 30, 5ft 10in to 6ft tall, slim and with a tanned complexion. He fired once, possibly using a sawn-off shotgun, inflicting fatal head wounds.
Welsh, who was jailed for running a protection racket and assaulting a woman, had a young daughter and a stepson. He ran Holyrood Boxing Gym and was involved in a number of charities, earning him praise for turning his life around after his criminal past.
However, he was closely linked to Edinburgh drugs baron Mark Richardson, one of Scotland’s most feared organised crime bosses, currently serving time for his role as the head of a drugs gang linked to multiple violent attacks.
In the 80s, he was involved in the Hibs football hooligan gang Capital City Service and fans at Easter Road yesterday held a minute’s applause in his memory..
He was jailed in 1990 for menacing an estate agent but turned to boxing after his release and in 1993 was crowned British ABA lightweight boxing champ but his career ended when he was jailed again, for four months, for assaulting a woman in her own home.
Meanwhile, police have denied suggestions Welsh died because of a missing £130,000 drugs consignment he was paid to protect.
They have also not ruled out a link to Somali gangs dealing crack cocaine in Edinburgh and elsewhere in Scotland.
Officers are still trawling CCTV in an attempt to identify the gunman, the route he used to and from the shooting scene, and if he was driven away by an accomplice parked in a nearby street.
A senior Police Scotland source said: “There are a number of things that can be taken as guaranteed in a case like this. Some one had a gripe and they acted on it. Whether that was local, Scotland-only or linked to gangs from England operating up here is yet to be determined.
“But you don’t get your money back if you kill the person in debt to you. That doesn’t not make any sense.”
He added: “They key is the description of the suspect and finding CCTV that shows him or suspect vehicles in the area.