The elite Counter Terrorism Service (CTS) pushed into the northeastern Sadeeq neighborhood, officers on the ground said.
A Reuters reporter in eastern Mosul said CTS forces were engaged in clashes in Sadeeq and were firing into neighboring Hadba, where their units had been fighting the day before.
Securing Hadba, Sadeeq and other nearby districts will allow the CTS to advance further toward the Tigris river that runs through the city, control of whose eastern bank will be crucial to launching attacks on western Mosul. Islamic State still holds all Mosul districts west of the river.
Forces also clashed with the militants further south, a military statement said, seeking to build on gains along the river bank, which they reached last week for the first time in the nearly 3-month campaign.
The U.S.-backed offensive to drive Islamic State out of Mosul, its last major Iraq stronghold, involves a 100,000-strong force of Iraqi troops, Kurdish fighters and Shi'ite militias.
The ultra-hardline Sunni group's loss of the northern city would probably spell the end of the Iraqi side of its self-styled caliphate, which spans parts of Iraq and Syria.