Mr Bradley, 59, was attacked in a Derry bar as he watched a televised football match with his son.
The former priest suffered head injuries and is in Altnagelvin Hospital overnight for observation.
Dissident republicans have been blamed, as they have petrol bombed Mr Bradley’s home and threatened him in the past.
But the latest attack was the most vicious yet and the detective leading the inquiry said it could have had fatal consequences.
Chief Superintendent Richard Russell said: “If you attack people in that way, hitting them on the head with implements of that kind, then they can be killed. There is no doubt about it.”
Northern Secretary Peter Hain and politicians on both sides of the Border condemned the attack but forecast it would not deter Mr Bradley from driving forward policing reforms.
He was targeted as he prepared to watch the second half of the Eircom League Cup final between Derry City and UCD.
It is understood police have taken statements from 16 individuals who were in the bar at the time. The bar does not have CCTV cameras and footage from a neighbouring off-licence has been of no use.
Chief Supt Russell said: “The man walked into the bar, approached Denis and hit him with something similar to a baseball bat.
“Denis was knocked down but we are not sure how many times he was struck. He suffered two lacerations to his head and a suspected broken nose.”
He said the suspect was wearing a light coloured hooded top and had a scarf around his face.
“We don’t think this is a random attack but it may have been an opportunistic attack. One of the major lines of inquiry is in relation to dissident republicans who have previously threatened Denis Bradley.”
One of the many who condemned the attack was Sinn Féin’s chief negotiator Martin McGuinness.
“The attack on Denis Bradley was wrong and unacceptable and must be condemned. The fact that he is the vice-chairperson of the Policing Board does not warrant an attack on either him or his family.”