A renewed push launched on March 5 has forced Daesh out of several neighborhoods and key sites, including the main local government headquarters and the famed Mosul museum, tightening the noose around fighters there.
West Mosul is the most-populated urban area still held by the terrorists, followed by Syria's Raqqa.
Iraq's Joint Operations Command announced additional gains on Monday, saying that forces from the elite Counter Terrorism Service had recaptured west Mosul's Al-Nafat neighborhood.
Lieutenant General Raed Shakir Jawdat said that forces from the Rapid Response Division, another special forces unit, and the federal police were working to search and clear territory on the edge of Mosul's Old City.
The forces are conducting "combing and search operations in the liberated areas of Bab al-Toub, searching for traps and mines and terrorists hiding among the people," Jawdat said in a statement, AFP reported.
Mosul's Old City – a warren of narrow streets and closely spaced buildings where hundreds of thousands of civilians are believed to still reside – could see some of the toughest fighting of the campaign to retake Iraq's second city.
While CTS and Rapid Response are spearheading the advance inside Mosul, Iraqi army forces and pro-government paramilitaries are fighting Daesh to the west.
Soldiers from the 9th Armored Division scored a key victory on Saturday night when they cut the last road out of west Mosul, said Brett McGurk, the US envoy to the so-called international anti-Daesh coalition.
"Any of the fighters who are left in Mosul, they're going to die there, because they're trapped," McGurk told journalists in Baghdad.
"We are very committed to not just defeating them in Mosul, but making sure these guys cannot escape," he said.
In practice, Daesh terrorists may still be able to sneak in and out of the city in small numbers, but the lack of access to roads makes larger-scale movement and resupply more difficult, if not impossible.
"We now believe that we are killing so many of their fighters that they are not able to replace them. That was not the case even a year ago," said McGurk, putting the toll for Daesh leaders at 180 dead.