Former Lord Advocate Dame Elish Angiolini completed an investigation into former practices at Mortonhall Crematorium and said in a statement it was delivered on the Monday before Easter.
Council chief executive Sue Bruce had said earlier tonight she would now be writing to the families to inform them the report was complete and to discuss the timing of its release.
However, many of the families were unaware of the long-awaited development before it was reported, according to their legal firm Thompsons.
Solicitor Patrick Mcguire, who represents most of the families involved for the legal firm, said they were stunned to learn of the report's completion in this way.
In a statement, Mr Mcguire said: "I would have to say that the families and myself are stunned that we learned this news through reports in the media.
"My clients are bereaved parents who have been to hell and back over this whole business and they expect to be treated in a sympathetic and decent manner.
"Being kept properly informed is the least they could expect from Edinburgh City Council after what they have been through. They do not expect to receive news like this from media reports."
He said they were 'extremely concerned' at any suggestion the report had been handed to the council last week. In a separate statement, Dame Elish confirmed she had delivered it on Monday, April 14.
Mr Mcguire added: "The parents and myself are also extremely concerned at any suggestion that the council have been in possession of this report since last week.
"They must come out publicly and state unequivocally when they received the report.
"If it transpires that they have been sitting on this for several days, it will constitute a huge breach of trust by the Council who have already behaved in a manner that has severely tested the trust of my clients."
Earlier, Ms Bruce said; "I have now received a copy of Dame Elish Angiolini's report.
"I fully appreciate that parents will be anxious to read the report and we are keen for them to receive a copy as soon as possible. We will be writing to parents in the coming days outlining the timing of publication.
"I would like to thank the families for their ongoing patience and Dame Elish and her team for their hard work on this important and sensitive investigation."
The scandal emerged in December 2012 when it was revealed that the crematorium had buried or scattered the ashes of babies for decades without their parents' knowledge.
Families said they were told there would be nothing to scatter.
Dame Elish said she had passed the report to the council chief executive on April 14. She thanked all parents and witnesses for their co-operation.
An independent commission, led by former high court judge Lord Bonomy, was set up to review policies and practice across Scotland in relation to the handling of ashes following the cremation of babies and infants.