Iraqi forces took the eastern side of Mosul from Daesh in January, after 100 days of fighting, and started attacking the western side in February.
Up to 350 militants are estimated by the Iraqi military to be besieged in the Old City, dug in among civilians in crumbling houses and making extensive use of booby traps, suicide bombers and sniper fire to slow down the troops' advance.
Assadi said Iraqi forces had linked up along al-Faruq, a main street bisecting the Old City, and would start pushing east, toward the river. "It will be the final episode," he said.
More than 50,000 civilians, about half the Old City's population, remain trapped behind Daesh lines with little food, water or medicines, according to those who escaped.
Aid organizations say Daesh has stopped many from leaving, using them as human shields. Hundreds of civilians fleeing the Old City have been killed in the past three weeks.
Daesh has carried out sporadic suicide bombings in parts of Mosul using sleeper cells. It launched a wave of such attacks late on Sunday, trying to take control of a district west of the Old City, Hay al-Tanak, and the nearby Yarmuk neighbourhood.
Social media carried posts showing black smoke and reports that it came from houses and cars set alight by the militants. Witnesses said residents had fled the two neighbourhoods.
Assadi said the attempt to take over the neighbourhoods had failed and the militants were now besieged in one or two pockets of Hay al-Tanak. A curfew was in force over western Mosul, a Reuters correspondent reported.
The fall of Mosul would mark the end of the Iraqi half of the "caliphate", but Daesh remains in control of large areas of both Iraq and Syria.
Baghdadi has left the fighting in Mosul to local commanders and has been assumed to be hiding in the Iraqi-Syrian border area. There has been no confirmation of Russian reports over the past days that he has been killed.
In Syria, the insurgents' "capital" Raqqa, is nearly encircled by a U.S.-backed, Kurdish-led coalition.
