The ferry was said to be heading towards a tourist island as part of new year celebrations.
Prime Minister Adel Abdul Mahdi has ordered an investigation "to determine responsibilities".
In a statement Mr Mahdi said he was following the story "with pain and sadness", and had ordered "all state efforts" to find survivors and treat victims.
Health and security sources said 72 people were killed, with 67 rescued and 55 still missing.
At least 55 people are reported to have died after a ferry sank in the Tigris river in the Iraqi city of Mosul.
The health ministry said 33 women, 12 children and 10 men had died, according to the state-run Iraqia News TV.
Most of the 100 passengers said to have been on board could not swim, the head of Mosul's civil defence agency said.
Rescue teams are trying to reach people who were on the ferry, which was reportedly heading towards a tourist island.
Authorities had reportedly warned people about rising water levels as the gates of the Mosul dam had been opened, and people are reportedly accusing the ship operator of ignoring the advice.
Images on social media show the upturned vessel and people floating in the fast-flowing river.
The vessel was reportedly on its way to Umm Rabaen island, a tourist area about 4km (2.5 miles) upstream.
Local information source Mosul Eye has tweeted an "urgent appeal" for people to go down to the river near the Fifth Bridge and the Old Bridge in the city centre and help with the rescue.
It called on motorists to stop their cars to allow ambulances carrying victims to pass.