Officials say the remains are "biomedical waste" from at least three hospitals in the Urals region, where they were discovered.
According to police, they were preserved with formaldehyde in four plastic barrels with tags listing surnames and numbers.
Villagers made the discovery in woodland about 70km (45 miles) north of the city of Yekaterinburg.
Police said some of the lids on the barrels had come off and the contents were scattered on the floor of the forest.
The health authorities say responsibility lies with the organisation charged with the disposal of medical waste from the city's hospitals.
"It seems the company responsible for disposal of the biomedical waste did not carry out its duties," Vladimir Vlasov, deputy head of the regional government, said on state television.
The remains have been placed in a morgue in the nearby town of Nevyansk for further examination.
'Little baby bodies'
The foetuses were found a few kilometres from a road linking Yekaterinburg with another big city, Nizhny Tagil.
They were discovered by a fisherman, local resident Sergei Tveritinov said.
"A friend called at night and said he had gone fishing and wanted to get some wood for his fire," Mr Tveritinov told state TV.