A senior official at the laboratory to which the bodies were taken said the people appeared to have died during the suppression of a Shia uprising against Saddam Hussein after the 1991 Gulf War.
The official said: "There are 31 bodies. We're still testing but it appears they are victims of the events of 1991."
There had been confusion over the scale of the mass grave found at a building site for a sewage project in the city centre, with police initially saying there were 150 bodies.
Rahman Mishawi, a police spokesman, later revised that to "dozens" of sets of remains.
A Reuters reporter at the scene saw one sack filled with what appeared to be human bones.
With Saddam on trial for crimes against humanity, the Shia-led government is keen to remind Iraqis of their suffering under his Sunni-dominated administration.
Some minority Sunni Arab leaders have accused police and other government officials of exaggerating accounts of atrocities and of using Saddam's trial for sectarian political advantage.
About 300 possible mass graves have been reported since Saddam's fall in April 2003, many in southern Shia areas of the country and in Kurdish areas of the north.
Human rights activists estimate that hundreds of thousands of people disappeared during Saddam's rule.