Elite Counter-Terrorism Service troops seized control of the state television building in Kukjali hours after launching an assault on the area.
But a BBC journalist embedded with them says they are facing fierce resistance.
Units of the army’s ninth division are meanwhile said to be bearing down on south-eastern districts of the city.
On Monday, Iraqi Prime Minister Haider al-Abadi told the 3,000 to 5,000 militants believed to be inside Mosul, which they overran in June 2014, that there was “no escape” and to “either surrender or die”.
About 50,000 Iraqi security forces personnel, Kurdish fighters, Sunni Arab tribesmen and Shia militiamen are involved in the two-week-old offensive to drive IS militants out of their last major urban stronghold in the country.
CTS units retook Bazwaya, the last village before Mosul’s eastern outskirts, in a dawn assault on Monday and then advanced on the Kukjali industrial zone.
They moved out again shortly before first light on Tuesday, this time with the aim of entering the adjoining Kukjali residential area, which is within the city limits.
If the troops can establish a foothold inside Mosul, it will be a big day for Iraq and all the other countries involved in the fight against the jihadist group, our correspondent says.
However, they are coming up against much more fierce resistance than he has seen in the past few days.
The troops have responded to the RPG, machine-gun and sniper fire with heavy weapons, and also by calling in air strikes by the US-led coalition against IS.
A number of civilians have also approached them, some waving white flags.
The Iraqi military announced on Tuesday that the ninth armoured division was approaching the south-eastern districts of Somer, Palestine, Yarimjah and Intisar, having recaptured the outlying villages of Allag, Tahrawah and Tubraq Ziyarah.