Garda sources said their appeal included the possibility that some cutting equipment may have been used over the weekend or left in a different position or setting.
They said it could be the case that equipment, or a room, was unexpectedly cleaned or partially cleaned.
Officers also urged people who are concerned as to the whereabouts of a person, but who have not reported the matter to gardaí, to do so straight away.
The fresh appeals were made as gardaí again asked people who were along the canal near Ardclough village, in Co Kildare, between last Thursday and Saturday afternoon to contact them.
The torso of a man, described as white and that of a young person, was found inside a large wheelie suitcase by passerbys at 3.30pm on Saturday.
Tests are ongoing on the torso, as well as on the suitcase and the wrapping the remains were found in.
Superintendent Gerry Wall, who is leading the investigation, said it was a “gruesome discovery” for the two people who pulled the suitcase out and opened it.
He said the investigation team knew the man was killed at another location.
“We are asking people returning to their property or premises after the weekend to check to see if there was anything unusual,” said Supt Wall, speaking at Leixlip Garda Station.
“Was a room or equipment disturbed or their premises in anyway different than when they left?
“Did you as an employee or employer notice anything different in your premises?”
He added that there may have been an “attempt to clean up” the room that the person hadn’t expected.
Separate Garda sources, while eager to respect the dignity of the victim, said the scene where the man was brutally assaulted and dismembered “had to be very messy” and that “out of the ordinary” efforts might have been taken to try and clean up the place.
“It had to happen somewhere and there has to be traces left of the crime, which is crucial for the investigation,” said one source.
The source said that if anyone is suspicious that something violent may have gone on at a premises, or in a room, to contact them.
Gardaí have already established that the man was killed in recent days, the latest being last Thursday.
This is thought to be based, in part, on the condition of the torso.
Supt Wall described the remains as that of a “young man”, but said they had no clues regarding nationality.
It is thought that no distinguishing tattoos or other marks are on the body.
DNA samples from the man have been sent to Forensic Science Ireland and Interpol to cross-reference their respective databases.
It takes a day to develop a DNA profile from a sample. FSI will run it against the DNA Database profile registry for any match. As it was only set up last November this is thought unlikely.
They will also run it against the crime scene index, which goes back much further.
The suitcase and the plastic wrapping the body was in will be subject to forensic analysis — including DNA, fingperprint and fibres — which will take a number of weeks. Any fingerprints found will be checked against the Garda Fingerprint Database.
The prospects of forensic samples are reduced given that the suitcase was in the water.
If gardaí develop any suspicions as to the identity they will seek to get a DNA sample from a family member to confirm.
Supt Wall said they had received some 80 calls from the public regarding their appeal for witnesses at Ardclough and said the information was being examined. This time-consuming process will hopefully give investigators a good idea when the suitcase was dumped into the water.
While officers have put the time-frame back to Thursday, sources suspect it was put there overnight on Friday.
The spot is some 500 metres from Ardclough Bridge, also known as Henry Bridge. It is outside the entrance to The Village, a restaurant in the grounds of the late Tony Ryan’s Lyons estate.
As it stands gardaí do not suspect the remains are those of anyone reported missing in recent weeks.
But they have urged people who have not reported someone who is missing, or who they are concerned about, to do so.
Contact Leixlip Garda Station on 01-6667800.
Workers and housemates urged to check if ‘a room or equipment’ was disturbed over the weekend