Intense shelling and gunfire rocked Rafah in southern Gaza on Thursday, after Israel declared it had seized a strategic corridor along the Palestinian territory's border with Egypt.
Israel launched its military incursion into Rafah in early May despite international objections over the safety of civilians sheltering in the city.
A strike that sparked a fire and killed dozens in a displacement camp at the weekend drew a wave of fresh condemnation.
Israel, which has repeatedly vowed to destroy Hamas after the Palestinian militant group attacked southern Israel on 7 October, said on Wednesday its forces had taken over the 14-kilometre (8.5-mile) Philadelphi corridor along the Gaza-Egypt border, which it alleges was used for weapons smuggling.
Military spokesman Daniel Hagari said Israel had taken "operational control" of the narrow border strip, where he said troops had "discovered around 20 tunnels".
Egypt, a longtime mediator in the conflict, has rejected claims of smuggling tunnels running beneath the buffer zone.
"Israel is using these allegations to justify continuing the operation on the Palestinian city of Rafah and prolonging the war for political purposes," a high-level Egyptian source was quoted as saying by state-linked Al-Qahera News.
Egyptian officials have said a potential Israeli takeover of Philadelphi could violate the two countries' 1979 peace deal, though there has been no official comment from Cairo since the military's announcement.
In Beijing, Egyptian president Abdel Fattah al-Sisi called for increased humanitarian assistance to Gaza, and reiterated his country's opposition to "any attempt at forcing Palestinians to flee their land".
His host, Chinese leader Xi Jinping, called for a "broad-based, authoritative and effective international peace conference" to address the war.
Hamas said later it had informed mediators that it would only agree a "comprehensive" truce agreement including a hostage-prisoner swap if Israel halts its "aggression".
On the ground in Gaza, witnesses reported fighting in central and western Rafah and bombardment in the east that killed at least five people near an aid warehouse.
AFP images from Al-Aqsa Martyrs Hospital in central Deir al-Balah showed children who appeared to be malnourished awaiting treatment.
The UN has warned of looming famine in Gaza. Israel said at the weekend that aid deliveries had been stepped up, including through its Kerem Shalom crossing with Gaza.
Witnesses said Israeli forces demolished buildings in east Rafah where the Israeli incursion began on 7 May, initially focusing on the vital Rafah border crossing, a key entry point for aid.
On Thursday, Israel said its army had killed about 300 Palestinian militants in Rafah since launching its operation in the city.
A stream of civilians fled Rafah, taking their belongings on their shoulders, in cars or on donkey-drawn carts.
Before the Rafah offensive began, the United Nations said up to 1.4 million people were sheltering in the city. Since then, one million have fled the area, the UN agency for Palestinian refugees, UNRWA, has said.
The Israeli military said it struck more than 50 targets across Gaza while its troops found weapons, explosives and tunnel shafts in Rafah, and battled militants in Jabalia.
Sunday's Israeli strike and resulting fire at the Rafah displacement camp killed 45 people, according to Gaza officials, and prompted two days of discussions at the UN Security Council.
Israel has said it targeted a Hamas compound and killed two senior members.
After the strike, Algeria presented a draft resolution to the UN Security Council demanding an immediate ceasefire and the release of all hostages, but it was unclear when it would be voted on.
Decisions by Spain, Norway and Ireland to formally recognise Palestinian statehood this week have sparked a debate over the issue.
Slovenia's parliament speaker said lawmakers in the former Yugoslav republic would vote Tuesday on whether to join the recognition moves. Israeli Foreign Minister Israel Katz said a yes vote would be a "reward" to Hamas.
The Gaza war was sparked by Hamas's 7 October attack on southern Israel, which resulted in the deaths of 1,189 people, mostly civilians, according to an AFP tally based on Israeli official figures.
Militants also took 252 hostages, 121 of whom remain in Gaza, including 37 the army says are dead.
Palestinian militant group Islamic Jihad released a second video of an Israeli hostage who was identified by Israeli media as Sasha Trupanov, 28.
Israel's retaliatory offensive has killed at least 36,224 people in Gaza, mostly civilians, according to the Hamas-run territory's health ministry.
Israel's National Security Adviser Tzachi Hanegbi said the war could go on until the year's end.
"We may have another seven months of fighting to consolidate our success and achieve what we have defined as the destruction of Hamas's power and military capabilities," Hanegbi said.
Amid the fighting, Israeli war cabinet minister Benny Gantz's centrist party submitted a bill to dissolve parliament for an early election, drawing criticism from Prime Minister Benjamin Netanyahu's right-wing Likud party.
The United States has urged Israel to refrain from a full-scale Rafah offensive because of the risk to civilians.
However, the White House said Tuesday that so far it had not seen Israel cross President Joe Biden's "red lines".
The Israeli seizure of the Rafah crossing has further slowed sporadic deliveries of aid for Gaza's 2.4 million people and effectively shuttered the territory's main exit point.
Cyprus, the European Union's easternmost member, said humanitarian aid shipped to Gaza was being kept at sea off the territory's coast, after a US-built pier was damaged in bad weather.