President Omar el-Bashir said he hoped "contacts that have helped us reach an understanding in many regional and international questions could be boosted," the Akhbar Al-Sa'a reported.
The newspaper did not say how the invitation was conveyed.
Last week, Mr. Carter said an investigation should determine whether the Aug. 20 attack was warranted on El Shifa factory. Washington said the plant produced precursors for chemical weapons.
"If the evidence shows that the Sudanese are guilty, they should be condemned for lying and for contributing to terrorist activities," Mr. Carter said Thursday. "Otherwise, we should admit our error and make amends to those who have suffered loss or injury."
The U.S. administration has so far turned down Sudanese calls for a U.N. fact-finding mission at the plant.