Al-Nujaify told Al Jazeera's Faultlines programme: "They entered Mosul illegally: two American citizens accompanied by an interpreter from Kurdistan province. They entered from the north.
"One of them holds a residence [permit] in Kurdistan. It is apparent he visited a number of Muslim countries," he said.
"We did not arrive at a conclusion in their interrogation. I believe they will ultimately be handed over to the American forces, since till now there is no evidence to claim they have committed a crime."
The Associated Press news agency quoted an unnamed offical as saying the Americans had been detained over suspected visa violations.
Al-Nujaify said they were arrested in an area considered "a hot zone".
"They were detained by one of the security patrols. They were not accompanied by any other persons. We do not know what their exact destination was," he said.
Josh Rushing, Al Jazeera's correspondent reporting from Mosul, said: "All this comes on the heels this week of a [US] senate foreign relations committee report that indicates that 36 Americans, who were ex-convicts, had travelled to Yemen in the past year, ostensibly to learn Arabic.
"Many of those have actually disappeared, where the government assumes they have gone to al-Qaeda training camps in the remote parts of the region."