The U.N. Humanitarian Coordinator for Sudan, Ameerah Haq, called for immediate access to the wounded.
"I am deeply perturbed by the reported bombings of a school, water installations and a market where civilians, especially women and children are present," she said in a statement.
Gen. Martin Luther Agwai, commander of the joint African Union-UN peacekeeping mission, said the bombings were "unacceptable acts against civilians" and said recent Darfur violence reflected a "total lack of commitment" by the government and the rebel groups to the peace process.
U.S.-based Darfur Diaries said six children, ages 4 to 11, were killed in an airstrike Sunday on a school it funds in the village of Shegeg Karo in North Darfur. Six more people were killed when the village's market area was bombed.
Residents reported that a Sudanese government aircraft hovered over the area for some time before repeatedly bombing it, the aid group said.
The U.S. State Department criticized the bombings and called for all armed groups to halt attacks on civilians.
"We condemn the aerial bombings by the Sudanese Armed Forces of civilian targets in North Darfur, including the reported May 4 attack on a primary school and marketplace which took the lives of several civilians," department spokesman Sean McCormack said.
There was no immediate government comment on the reported attacks, which could not be independently verified. Sudanese forces frequently bomb rebel areas in Darfur, despite a U.N. Security Council ban on military flights over the western region.
Some 35 people were reportedly injured, including two children in critical condition, according to Darfur Diaries.
To evacuate the wounded, troops need to get permits, coordinate with the Sudanese government and aid agencies in the area, and find government, accusing it of discrimination.
Sudan denies backing the janjaweed militia of Arab nomads accused of the worst atrocities in the conflict.
