Advances in recent days, including Saturday's push to within several hundred meters of the Tigris, have included an unprecedented nighttime assault by elite forces and driven the militants out of several areas east of the river in what is their last major stronghold in Iraq.
Counter-terrorism service (CTS) forces were the closest they had been to the Tigris inside Mosul and closing in on a strategic bridge, the spokesman said. The operation to recapture the city was launched nearly three months ago.
"Counterterrorism forces have been sent about 500m (yards) from the fourth bridge," Sabah al-Numan told reporters east of Mosul.
CTS seized the Ghufran district, previously known as al-Baath, and entered neighboring Wahda, he said, AFP reported.
A separate military statement said Iraqi federal police had recaptured a hospital complex in Wahda in southeastern Mosul, a significant turnaround after army units were forced to withdraw from the site last month.
Numan said fresh advances, which have gathered pace after troops were bogged down for several weeks by the presence of large numbers of civilians, were a consequence of new tactics and better coordination between different branches of the military.
CTS and federal police "are now moving in parallel on both axes" in southeastern Mosul, he said.
"We are proceeding side by side ... and advancing at the same level. This is a very important factor, thanks to which Daesh has not been able to move its fighters. It has to support one axis (front) at the expense of another.
"We have worn down the terrorist organization with this type of advance."
Friday's nighttime operation, launched after a week of planning, had been a particular success, Numan said.
CTS forces using night-vision equipment crossed the Khosr river, a tributary that runs perpendicular to the Tigris through eastern Mosul, via makeshift earth bridges after Daesh had destroyed permanent ones, he said.
The CTS and federal police are part of a 100,000-strong Iraqi force made up of the military, Kurdish fighters and popular forces.