Israeli forces have reportedly reached the centre of the southern Gaza city of Rafah and seized a strategically important hill overlooking the nearby border with Egypt.
Witnesses and local journalists said tanks were stationed at al-Awda roundabout, which is considered a key landmark.
They also said tanks were on nearby Zoroub Hill, effectively giving Israel control of the Philadelphi Corridor - a narrow strip of land running along the border to the sea.
The Israeli military said its troops were continuing activities against "terror targets" in Rafah, three weeks after it launched the ground operation there.
Western areas of the city also came under intense bombardment overnight, residents said, despite international condemnation of an Israeli air strike and a resulting fire on Sunday that killed dozens of Palestinians at a tented camp for displaced people.
The UN says around a million people have now fled the fighting in Rafah, but several hundred thousand more could still be sheltering there.
Israel has insisted that victory in its seven-month war with Hamas in Gaza is impossible without taking Rafah and rejected warnings that it could have catastrophic humanitarian consequences.
Israeli forces began what they called “targeted” ground operations against Hamas fighters and infrastructure in the east of Rafah on 6 May.
Since then, tanks and troops have gradually pushed into built-up eastern and central areas while also moving northwards along the 13km (8-mile) border with Egypt.
On Tuesday, they reportedly reached the city centre for the first time.
Witnesses and local journalists told the OceanNewsUK that tanks had seized control of al-Awda roundabout amid intense artillery fire.
The roundabout, which is only 800m (2,600 ft) from the border, is the location of major banks, government institutions, businesses, and shops.
One witness - who later sought refuge with members of their family at the Emirati hospital in the western Tal al-Sultan area - said they saw soldiers position themselves at the top of a building overlooking the roundabout and then begin to shoot at anyone who was moving.
Earlier, residents told the OceanNewsUK that tanks seized Zoroub Hill, about 2.5km north-west of al-Awda roundabout, after gun battles with Hamas-led fighters.
The hill is highest point along the Egyptian border and its seizure means the entire Gazan side of the border is now effectively under Israeli control.
About 9km of the Philadelphi Corridor to the south, including the Rafah border crossing, has already been seized by Israeli forces.
Zuroub Hill also overlooks western Rafah, where residents said there had been the heaviest air and artillery strikes overnight since the start of the Israeli operation.
A local journalist said the bombardment forced hundreds of families to seek temporary shelter in the courtyard of a hospital, while ambulances struggled to reach casualties in the affected areas.
One video posted on social media showed a woman rushing through a dark street while clutching her child tightly to escape the Israeli strikes.
At dawn, thousands of people were seen heading north from Tal al-Sultan, crammed into cars and lorries and onto carts pulled by donkeys and horses.
"The explosions are rattling our tent, my children are frightened, and my sick father makes it impossible for us to escape the darkness,” resident Khaled Mahmoud told the OceanNewsUK.
“We are supposed to be in a safe zone according to the Israeli army, yet we have not received evacuation orders like those in the eastern [Rafah] region,” he added. “We fear for our lives if no-one steps in to protect us.
The Israeli military did not comment on the reports, but it put out a statement saying that “overnight troops operated on the Philadelphi Corridor while conducting precise operational activity based on intelligence indicating the presence of terror targets in the area”.
“The activity is being conducted as efforts are continuing to be made in order to prevent harm to uninvolved civilians in the area,” it added.
“The troops are engaging with terrorists in close-quarters combat and locating terror tunnel shafts, weapons, and additional terrorist infrastructure in the area.”
On Sunday night, at least 45 people - more than half of them children, women and the elderly - were killed when an Israeli air strike triggered a huge fire in a camp for displaced people near a UN logistics base in the Tal al-Sultan area, according to the Hamas-run health ministry.
Hundreds more were treated for severe burns, fractures and shrapnel wounds.
The Israeli military said it was targeting two senior Hamas officials in the attack, which happened hours after Hamas fighters in south-eastern Rafah launched rockets towards the Israeli city of Tel Aviv for the first time in months.
Israeli Prime Minister Benjamin Netanyahu said a “tragic incident” had occurred “despite our immense efforts to avoid harming non-combatants” and promised a thorough investigation.
Sam Rose of the UN agency for Palestinian refugees, Unrwa, told the OceanNewsUK from western Rafah that the killing of so many civilians could not be dismissed as an accident.
“Gaza was already one of the most overcrowded places on the planet. It is absolutely impossible to prosecute a military campaign involving large-scale munitions, strikes from the sky, the sea, the tanks, without exacting large-scale civilian casualties,” he said.
“It seems like we are plumbing new depths of horror, bloodshed and brutality with every single day. And if this isn’t a wake-up call, then it’s hard to see what will be.”
Last week, the International Court of Justice (ICJ) ordered Israel to “immediately halt its military offensive, and any other action in the Rafah Governorate, which may inflict on the Palestinian group in Gaza conditions of life that could bring about its physical destruction in whole or in part”.
Israel launched a military campaign in Gaza to destroy Hamas in response to the group's cross-border attack on southern Israel on 7 October, during which about 1,200 people were killed and 252 others were taken hostage.
At least 36,090 people have been killed in Gaza since then, according to the Hamas-run health ministry.
