It was from this medieval mosque that the militants' leader Abu Bakr al-Baghdadi declared a self-styled ''caliphate'' spanning parts of Syria and Iraq three years ago.
ISIS's Amaq news agency accused American aircraft of destroying the mosque.
"The Daesh (ISIS) terror gangs committed another historical crime by blowing up the al-Nuri mosque and its historical al-Hadba minaret," the Iraqi military statement said, .
The explosions happened as Iraq's elite Counter Terrorism Service units, which have been battling their way through Mosul's Old City, got to within 50 meters (164 ft) of the mosque, the statement said.
Iraqi forces earlier on Wednesday said they had started a push towards the mosque. A U.S.-led coalition is providing air and ground support to the Mosul offensive which began in October 2016.
Al-Baghdadi proclaimed himself "caliph", or ruler of all Muslims, from the pulpit of the mosque on July 4, 2014, after the insurgents overran parts of Iraq and Syria. His black flag had been flying over its leaning minaret since June 2014.
Iraqi officials had privately expressed the hope that the mosque could be captured in time for Eid al-Fitr, the festival marking the end of Ramadan, the Muslim fasting month. The first day of the Eid falls this year on June 25 or 26 in Iraq.
The fall of Mosul would, in effect, mark the end of the Iraqi half of the "caliphate" even though ISIS would continue to control territory west and south of the city, the largest they had control of in both Iraq and Syria.
Baghdadi has left the fighting in Mosul to local commanders and is believed to be hiding in the border area between Iraq and Syria, according to U.S. and Iraqi military sources.