Mortonhall Crematorium was said to have told a funeral director that it would have refused to cremate the body of a child sex offender, had the person's identity come to light.
In letters revealed through Freedom of Information laws, the council-run facility in Edinburgh was accused of claiming the "cremator would now need to be deep cleaned before and after the cremation".
The controversy comes two years ago after Glasgow City Council refused to cremate the body of Moors Murderer Ian Brady, 79, who had requested that his ashes were scatted in the River Clyde, near the Gorbals, where he was born.
The local authority said at the time it would reject the request, and would 'advise the private crematoria not to accept the request'.
Child killer Brady was eventually cremated in Southport, Merseyside, with no music or flowers, and his ashes were disposed of at sea.
But council-run crematoriums have now been told by The National Association of Funeral Directors (NAFD) that they cannot take exception to a person's crimes, due to the distress caused to relatives.
The NAFD has objected to such refusals, and said it would 'cause distress to an innocent family'.
A letter from its chief executive Jon Levett, to the City of Edinburgh Council acknowledged that cremations could take place in secret, if there was a risk of disorder.
Mr Levett wrote: "If Mortonhall Crematorium continues to adopt this policy, this could cause unnecessary problems for funeral directors and distress for the families they serve.
"We appreciate that in certain exceptional circumstances, where there is a significant risk of public unrest, it may be necessary to arrange for disposals to take place in secret.
"However, it cannot be right for a local authority to take it upon itself to pass judgement on people after death, no matter how we view their activities in life.
"We would also question whether a policy of refusing to cremate certain people is compatible with the local authority's public health responsibilities.
"Forcing a family to approach multiple crematoria before reluctantly agreeing to cremate a person will simply cause distress for the innocent family."
Mr Levett said an Edinburgh-based funeral director had taken instructions from the family of a sex offender, and a service and cremation had taken place.
However, he said Mortonhall had contacted the director prior to the funeral to say it "would have refused the cremation" had it known whose remains it was dealing with - and that the
"cremator would now need to be deep cleaned before and after the cremation".
Responding to the letter, the City of Edinburgh Council's chief executive Andrew Kerr, said in 'very exceptional' circumstances, a request could be rejected.
Mr Kerr wrote: "Depending on the particular factual circumstances, where there is a genuine concern about public unrest, the council may accept an application subject to conditions or, very exceptionally, refuse an application.
"Where there is concern that special arrangements may be required, council officers consult the Scottish Government's inspector of crematoria.
"It may be necessary to consider special arrangements such as cremation out of hours, or scattering or interment of ashes in a discreet area.
"In highly exceptional cases it may not be possible to grant an application for cremation.
"As noted above, there is no policy in this regard and each case is carefully considered on its merits."
Earlier this year Robert Swanson, Scotland's inspector of crematoria, revealed he was contacted "following concerns about potential reputational damage, and procedures to be carried out should a crematorium accept application for cremation of a high-profile criminal offender (eg child sex offender)".
But he said 'death is a certainty'.
A spokesman for the City of Edinburgh Council said: "We review applications for burial or cremation of high profile criminal offenders as they arise on a case by case basis."