Lothian and Borders Police said they had been contacted after the scandal broke surrounding Mortonhall Crematorium in Edinburgh and its treatment of cremated remains of stillborn babies and infants who survived only a few days.
Lawyers believe more parents will come forward with concerns about their babies' remains after being told there were none after .
So far 150 families have complained to Edinburgh City Council and 55 have contacted lawyers. Compensation is a key issue that will be explored.
Lothian and Borders Police said: "Both complaints will be thoroughly assessed to determine whether any further action is appropriate."
It comes after The Herald revealed practices for disposing of the remains of children who have been cremated vary across Scotland. A council inquiry into Mortonhall is to be led by an independent person but campaigners believe a full public inquiry is needed.
Thompsons partner Patrick McGuire, said more parents were expected to come forward that an inquiry was essential and criminal proceedings may follow.
Councillors have asked auditors from PricewaterhouseCoopers to look through 100,000 records.