"Sixty-five to 70 percent of the Old City has been liberated, there is less than a square kilometre left to retake", said Obeidi, from the elite Counter-Terrorism Service (CTS) that has spearheaded the assault.
A "couple of hundred" ISIS fighters still defend the Old City in western Mosul, though it's not clear how many civilians are trapped in the area, a US-led coalition spokesman said.
A US-led global coalition is providing air and ground support in the 8-month-old offensive.
'As our heroic forces are closer to declaring final victory over the Daesh (Islamic State) gangs, I offer my most honest congratulations for Eid al-Fitr, ' Iraq Prime Minister Haider al-Abadi said in a statement.
The mosque was also hugely symbolic - it was from a pulpit there that the Islamic State group's top leader Abu Bakr al-Baghdadi in July 2014 declared a self-styled "caliphate", encompassing territories held by IS in Syria and Iraq.
More than 50,000 civilians, about half the Old City's population, remain trapped behind ISIL lines with little food, water or medicines, according to those who escaped.
Iraqi troops on Monday captured Al Faruq quarter, facing the mosque, the military said. "It will be the final episode", he said.
Iraqi forces, backed by worldwide coalition, launched their final push on June 18 to drive out IS militants from al-Shifaa neighborhood and the densely-populated old city center.
Iraqi army soldiers and volunteer fighters from the Popular Mobilization Units, commonly known by their Arabic name Hashd al-Sha'abi, have made sweeping gains against Daesh since launching the Mosul operation on October 17, 2016.
Daesh has carried out sporadic suicide bombings in parts of Mosul using sleeper cells.
Maj Gen Rupert Jones, the British deputy commander of the global coalition fighting against Isil, said those left would be "cleared" by local forces.
A spokesman for the US -led coalition rejected the claim.
In Jakarta, residents said they hoped the spirit of Eid would overcome fears about rising militancy in the country with the largest number of Muslims, reports The Dawn.
From those neighbourhoods, they were bearing down on Raqqa's Old City in a pincer movement yesterday, with fighting raging in the western Al-Qadisiya district and parts of the city's east.
An Iraqi officer says counterattacks by Islamic State militants on the western edge of Mosul have stalled Iraqi forces' push in the Old City - the last IS stronghold in the city. Both were killed when the terrorist detonated his belt but no other casualties were reported. Modernization initiatives and long neglect had caused significant damage before Islamic State took over, yet the 150-foot (45-metre) leaning Hadba minaret had survived as an icon of the city.