Iraqi Prime Minister Haider al-Abadi, who is also the commander-in-chief of the armed forces, announced the start of an offensive on February 19 to drive extremist militants out of the western side of Mosul, locally known as the right bank of Tigris River, which bisects the city.
Federal police and Rapid Response units said they had entered the Bab al-Tob area of the Old City, where the fight is expected to be toughest due to narrow alleyways through which armored vehicles can not pass.
Mosul is by far the largest city which Islamic State has held in its cross-border, self-declared caliphate in Iraq and Syria.
A Sunni politician in Baghdad says Americans raised the Kurds, Iran raised the Shias, but we, Sunnis, are like abused children.
Mr McGurk said they were only six miles from the city. Reuters was not yet able to access the museum to verify.
The militants corralled the inmates, who had been serving sentences for a range of crimes - from murder and assaults to nonviolent offences - onto trucks and drove them to an isolated stretch of desert about 2 kilometres (1 miles) from the prison, HRW said.
"Our forces have seized control of 65 percent of western Mosul", he told Anadolu Agency.
"Mosul's liberation is increasingly in sight albeit with increasingly hard fighting ahead", McGurk said, adding that Iraqi forces were retaking "some of the most hard ground that we knew would have to be reclaimed".
Progress was slowed by heavy rain but units were only 100 metres from the Iron Bridge that connects the Old City with the eastern side of Mosul, military officials said.
Iraqi troops fire artillery towards Islamic State (IS) jihadists' positions in west Mosul.
USA officials believe that around half the 5,000 ISIS fighters in Mosul at the start of the offensive last October were killed or severely wounded.
Iraqi forces battling ISIL faced tough resistance from snipers and mortar rounds on Monday as they tried to advance on Mosul's Old City and a bridge across the Tigris River in their campaign to retake it.
The jihadist group has lost most of the cities it captured in northern and western Iraq in 2014 and 2015.
McGurk also announced that Iraqi Prime Minister Haider al-Abadi will visit Washington next week to hold discussions with United States president Donald Trump on the further cooperation between two countries. "If there are civilians, families in the homes, we shout to them to take cover inside a room". In Syria, it still holds Raqqa city as its stronghold, as well as most of Deir al-Zor province.
The US-backed Syrian Democratic Forces, an umbrella group for Kurdish and Arab fighters, are closing in on Raqqa.