The 33-year-old was discovered at the entrance to the Tesco Express on Quinsboro Road in the town just before 7am on Saturday.
"Preliminary indications are that the deceased passed away as a result of hypothermia," explained a source.
Emergency services were called to the scene where they attempted to revive him.
But according to reports any attempts to revive the victim failed as he had suffered a cardiac arrest.
He was pronounced dead shortly after being taken to Loughlinstown Hospital.
CCTV images showed that the man had been sleeping in the door of the supermarket prior to his death.
"It was an extremely cold night and investigators have established that he spent the night sleeping in the position where he was discovered by a supermarket staff member the next morning.
"It's a particularly tragic case," added the source.
Mayor of Bray Mick Glynn described the coastal town's shock at learning of the nature of Mr Doyle's death.
Mr Glynn has asked members of the public who come across people sleeping rough to urge them to seek help from the authorities.
"There is a huge problem with people sleeping rough, not just in Bray but in Dublin and all over the country," said the councillor.
"But there are services out there to help them if people accept the help that is being offered.
"I would urge members of the public who are in contact with people to try to convince them to seek the help that is there," he added.
"This is a very sad situation and hopefully one we can avoid in the future as there is emergency accommodation out there if help is sought."
While the man had been sleeping rough, sources say that the victim's family were completely supportive and had made every attempt to help him.
"Although he was sleeping rough on Friday night, his family was always there for him but there were personal issues at play that meant he didn't always accept that help," the source added.
"Paul comes from a large, loving and highly respected family in Bray who have done everything in their power to help him."