Intelligence Minister Haidar Moslehi said last month Tehran had proof that the three Americans, detained on espionage charges, had links to intelligence services.
Shane Bauer, 27, Sarah Shourd, 31, and Josh Fattal, 27, were detained after they entered Iran from northern Iraq, a case that has further complicated poor relations between Tehran and Washington.
"It was decided... to issue visas for these three mothers so they can come to Iran and visit their children," the semi-official Fars news agency quoted Mottaki as saying.
Mottaki said the decision to allow the visit was a "humanitarian" in nature, adding that the hikers' mothers could get their visas from the Iranian diplomatic mission in New York.
Iran's English-language Press TV had said earlier on Monday that the families of the detained Americans would be permitted to visit them, adding no date for the trip had been set.
In the United States, the families said they had not received official confirmation of the news.
"But we would obviously be delighted if it is true and are ready to travel as soon as our visas are issued. We are extremely concerned for the physical and emotional welfare of Shane, Sarah and Josh and cannot wait to see them after nine long months," they said in a joint statement.
In Washington, State Department spokesman P.J. Crowley said: "We have been seeking and supporting for some time efforts by the families of the three hikers to visit their loved ones in Tehran. We have communicated that to Tehran."
In a brief meeting with Mottaki last week, U.S. Deputy Ambassador to the United Nations Alejandro Wolff handed over letters from the families of the detained Americans.
Their families have said they were hiking and accidentally strayed across the border from Iraq. The White House has called for their immediate release. Under Iran's Islamic law, espionage can be punishable by death.
The United States has no diplomatic relations with Iran and is at loggerheads with the Islamic state over its disputed nuclear energy program.
Washington is lobbying UN Security Council members to back a fourth round of sanctions on Tehran, to press it into curbing sensitive atomic activity the West suspects is aimed at developing nuclear weapons ability. Iran denies the charge.
The United States and Israel, Iran's arch-foes, have not ruled out military action if diplomacy fails to resolve the row.
