The official, who spoke on condition of anonymity because the investigation is still under way, says the man apparently had two guns.
The shooting was reported at Sandy Hook Elementary School in Newtown, in western Connecticut. State police say Newtown police called them at about 9:40 a.m. about the reports.
Newtown is in northern Fairfield County, about 45 miles southwest of Hartford and 80 miles northeast of New York City.
Eight-year-old Alexis Wasik, a third-grader at the school, said police were checking everybody inside the school before they were escorted to the firehouse.
"We had to walk with a partner," she said.
One child leaving the school said that there was shattered glass everywhere. A police officer ran into the classroom and told them to run outside and keep going until the reach the firehouse.
Earlier reports of a second shooter are unconfirmed.
Associated Press, Reuters and the Hartford Courant contributed
Connecticut elementary school shooting: Multiple deaths reported
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Connecticut elementary school shooting: Multiple deaths reported
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Connecticut elementary school shooting: Multiple deaths reported