The first blast was near a local government building and the second one about 60 meters (65 yards) away at a bus station, police sources said.
Turmoil in Iraq: Blast along Baghdad shrine route kills 23
Unverified online photographs showed a large plume of smoke rising above the buildings as well as burnt out cars and bodies on the ground at the site of one of the blasts, including several children. Police and fire-fighters carried victims on stretchers and in their arms.
Islamic State holds positions mostly in Sunni areas of the country's north and west, far from the mainly Shia southern provinces where Samawa is located. Such attacks are relatively rare.
US military softens claims on drop in Islamic State's foreign fighters
The rise of the ultra-hardline Sunni insurgents has exacerbated Iraq's sectarian conflict, mostly between Shias and Sunnis, which emerged after the 2003 US-led invasion that toppled Saddam Hussein.
The quota-based governing system put in place by the United States at the time is being challenged by hundreds of protesters who camped out overnight in Baghdad's heavily fortified Green Zone after storming the parliament building.
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