The Iraqi army's ninth division and Shi'ite paramilitary forces said on Wednesday they had cut the main road between the city and the Islamic State stronghold of Tal Afar to the west, tightening a noose around the city.
CTS troops stormed the al-Jadida and al-Aghawat districts on Sunday, Major General Maan al-Saadi told reporters in Mosul, saying the militants were showing signs of weakness despite initial "fierce" resistance.
The United States has said the looting and smuggling has been a significant source of income for the militants.
The IS is facing simultaneous offensives by the government forces, Turkish-backed rebels and a US-supported alliance of Kurdish and Arab fighters, in western Mosul.
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"The enemy fought fiercely in the first defensive line from Al-Ghazlani base to Wadi Hajar and to Al-Sumood neighborhood", Saadi said, referring to areas that Iraqi forces have recaptured since unleashing their offensive on February 19.
The Iraqi authorities said they had no information at their disposal regarding the possible use of chemical weapons.
Last week, the troops reached the first bridge from the south in western Mosul, known locally as the 4th Bridge.
The 100,000-strong Iraqi force fully captured the eastern half of Mosul in January, and commanders began an operation to cross the Tigris and take the western half last month.
Iraqi forces have uncovered a mass grave of about 500 executed prisoners after they forced Isis out of a prison complex near Mosul.
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It comes as Iraqi government forces are battling to retake the northern city of Mosul from the Sunni extremist group.
These shocking pictures show the ravaged remains of Mosul's museum which has been looted and destroyed by evil Islamic State jihadis.
Mr Abadi said he would "not hesitate to strike the positions of terrorists", but stressed that attacks would only take place if the countries gave their consent.
Meanwhile, Iyad Rafed of the Iraqi Red Crescent Society said the Hammam al-Alil displacement camp - which accommodates up to 4,000 families in southeastern Mosul - is no longer able to receive any more refugees.
Iraqi forces aim to dislodge Islamic State militants from west Mosul within a month, despite grueling urban combat in densely populated terrain, the head of the elite Counter Terrorism Service told Reuters on Thursday. They then set their sights on the west side of the city. "They teach us more about Assyrian art".
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