The areas mentioned by Rasool - which are located north of Mosul's Old City, where IS also still holds significant territory - are the main targets of the offensive, which was announced on Saturday.
Military officer Jabbar Hassan, for his part, said Iraqi forces stormed Al-Shifa district amid heavy exchanges of fire with ISIL militants.
Thousands of Iraqi security forces, Kurdish Peshmerga fighters, Sunni Arab tribesmen and Shia militiamen, assisted by US-led coalition warplanes and military advisers, are involved in the offensive.
Iraqi forces took control of eastern Mosul in January and launched the battle in the west in February.
An Iraqi official said a senior Iranian Revolutionary Guard commander was killed in an explosion during clashes with the Islamic State group west of Mosul.
Mosul's Old City is an ancient district of narrow alleyways and tightly packed homes where the United Nations estimates more than 100,000 people are being held by IS. But after the federal police stalled on the southern edge of the Old City just weeks into the push, the joint command center adjusted the plans, ordering Iraqi forces to first sweep up and around the congested Old City.
Following the launch of the US investigation into the incident in late March, Mosul advances ground to a near halt for weeks.
A number of other houses in the area were also destroyed by clashes between IS fighters and US -backed Iraqi forces around the same time, according to residents interviewed by The Associated Press, but the Pentagon investigation looked into the single March 17 airstrike that hit at 8:24 a.m. local time.
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There have also been reports that members of an Iraqi interior ministry special forces unit tortured and killed detainees during the Mosul operation.
"Another reason is the presence of thousands of civilians under siege", he said.
"The Iraqi government must ensure all exit corridors are genuinely safe for people to flee", it said.
IRAQI forces faced fierce resistance from Isis extremists yesterday on the second day of the final push to liberate Mosul. Mosul residents have been suffering from food shortages for months, a hardship expected to be exacerbated during the holy month of Ramadan that began Friday night, when Muslims fast during daylight hours.
Save the Children warned that fleeing civilians could be caught in the crossfire, leading to "deadly chaos".
Also Saturday, May 27, 2017, aid groups said they are concerned for the safety of civilians following calls from Iraq's government for residents of the Islamic State-held Old City to flee the area immediately.
Civil protection rescue teams reported recovering more than 200 bodies from the area in the days following the March 17 strike.
According to a military investigation, both Iraqi forces and USA military advisors did not know there were some many people in the building near an IS target that collapsed as a result of the strikes.