Forces from Iraq's Counter-Terrorism Service (CTS) were fighting IS in Gogjali, a village on the eastern edge of Mosul that they reached yesterday.
While CTS may soon enter Mosul, which IS overran two years ago, it will likely have to wait for other units to catch up before making a concerted push into the city.
"We are currently advancing in Gogjali," Staff Lieutenant General Abdelwahab al-Saadi, a senior CTS officer, told AFP.
For CTS's "Mosul Regiment", which has been fighting its way towards the city, retaking it is a matter of pride.
They were the last to retreat when IS seized the city on June 10, 2014 and they want to be the first back in.
The voice of Colonel Mustafa al-Obeidi came sputtering over the radio as his men advanced cautiously through Bazwaya, a village east of Gogjali, on Monday, sidling along walls and scanning the empty streets with their rifles raised.
Backed by air and ground support from a US-led coalition, tens of thousands of Iraqi fighters are converging on Mosul on different fronts, in the country's biggest military operation in years.
Prime Minister Haider al-Abadi appeared on state television yesterday wearing camouflage uniform.
"We will close in on (IS) from every place," he said.
"They don't have an exit, they don't have an escape, they can only surrender - they can die or they can surrender."
For the time being, the jihadists do have an escape route - to the west towards IS-controlled territory in neighbouring Syria.
They are not directly headed for Mosul, instead setting their sights on the town of Tal Afar which commands the city's western approaches.
The Hashed said yesterday that they had retaken a series of villages during their advance and surrounded others.
Their leadership says publicly that they do not intend to enter Mosul, which has an overwhelmingly Sunni population, but commanders on the ground say they want to fight inside the city.
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