Israeli tanks backed by warplanes and drones advanced deeper into the western part of the Gaza Strip city of Rafah on Wednesday.
Israeli forces have laid waste to much of Gaza and seized most of the Palestinian territory but have yet to achieve Israel's stated goal of wiping out Hamas and freeing Israeli hostages.
An Israeli commander briefing military correspondents in Rafah on Tuesday named two more locations there - Shaboura and Tel Al-Sultan - where the army planned to take on Hamas fighters.
"The Hamas battalions there are not yet well worn down and we need to dismantle them completely. We estimate it at more or less a month, at this intensity," Colonel Liron Batito, head of the Givati Brigade, told Army Radio.
The Israeli military remained in control of the border between Rafah and Egypt. Footage circulated on social media showed the Rafah crossing, the only window for most of Gaza's population with the outside world, was destroyed, buildings burnt, and Israeli tanks positioned there with the flag of Israel flying over some places.
The Israeli military said aid into Gaza had not been impeded by the damage.
Further north, Israel sent a column of tanks back into the Zeitoun neighbourhood in Gaza City, and residents reported heavy fire from tanks and warplanes but also sounds of gun battles with Hamas-led fighters.
The armed wings of Hamas and Islamic Jihad said fighters battled Israeli forces with anti-tank rockets and mortar bombs, and have in some areas detonated pre-planted explosive devices against army units.
Later on Wednesday, Palestinian gunmen fired rockets at the Kerem Shalom crossing in southern Gaza, the Israeli military said.