The Palestine Red Crescent Society (PRCS) charity says the Gazan city of Rafah has come under "violent" attack, with a number of deaths reported.
Witnesses spoke of dozens of air strikes on the city's north and centre.
The Israeli military said it had conducted a strikes in southern Gaza, without giving further details.
It follows warnings from the international community over Israel's planned offensive in the city, where 1.5 million people are sheltering.
The United Nations has said there is nowhere safe to go for the Palestinians who have ended up in the far south of the Palestinian territory.
On Sunday, US President Joe Biden told Israel's prime minister a military operation in Rafah should not happen without measures to ensure the safety of civilians.
In a call with Benjamin Netanyahu, Mr Biden said Israel needed a "credible and executable plan" to protect the more than a million people in the city, according to the White House.
Mr Netanyahu has insisted it will go ahead and a plan is being prepared.
The call between the two leaders comes days after Mr Biden suggested Israel's military operations in Gaza were "over the top".
It also follows a raft of Israel's allies, international organisations and regional powers expressing growing concern at suggestions Israeli troops would enter Rafah - which lies on the border with Egypt and is the only open point of entry for humanitarian aid.
UK Foreign Secretary David Cameron said "over half of Gaza's population are sheltering in the area", while Saudi Arabia warned of "very serious repercussions" if Rafah was stormed.
Many of the people are living in tents in refugee camps, having already been forced to flee their homes elsewhere in Gaza at least once on the orders of the Israeli military.
Meanwhile, the Strip's Hamas rulers said there could be "tens of thousands" of casualties, warning that any operation would also undermine talks about a possible release of Israeli hostages held in the strip.
The warnings about an escalation of fighting in Rafah came after Mr Netanyahu ordered his military to prepare to evacuate civilians from the city ahead of an expanded offensive against Hamas.
Israel launched its operations in the Palestinian enclave after more than 1,200 people were killed in southern Israel on 7 October by Hamas gunmen, who also took about 240 people hostage.
On Sunday, the Hamas-run health ministry in Gaza said 112 more Palestinians had been killed by the Israeli military over the previous day, bringing the overall death toll to more than 28,100 and more than 67,500 injured.
In their call on Sunday, the White House said Mr Biden "reaffirmed his view that a military operation in Rafah should not proceed without a credible and executable plan for ensuring the safety of and support for the more than one million people sheltering there".
He repeated Israel and the US shared goal of seeing Hamas defeated and ensuring Israel's long-term security, while he also called for "urgent and specific steps" to increase humanitarian aid to civilians trapped in Gaza.
Negotiators working on a deal to secure the release of the remaining hostages held by Hamas in Gaza have made "real progress" over the last few weeks, a senior White House official told Reuters.
The official said this deal was the main focus of the call between the Israeli and US leaders, but they also told the news agency that there were still some significant gaps to close.
Mr Netanyahu rejected Hamas's proposed ceasefire terms last week.
In an interview with US broadcaster ABC News aired on Sunday, the Israeli PM said "victory is in reach" and the Israeli military were "going to get the remaining Hamas terrorist battalions in Rafah".
He also said Israel would "provide safe passage" for civilians in the southern city.
When pressed about where they should go, Mr Netanyahu suggested there were "plenty" of areas "that we've cleared north of Rafah" and insisted officials were "working out a detailed plan".
"Those who say that under no circumstances should we enter Rafah are basically saying, 'Lose the war. Keep Hamas there,'" he added.
The US has already warned Israel that an invasion of Rafah as part of its assault on Gaza would be a "disaster", while the EU and the UN both expressed their own concerns.
Aid groups say it is not possible to evacuate everyone from the city.
UN humanitarian co-ordinator Jamie McGoldrick, who has just been to Gaza to assess the situation, told the BBC's Barbara Plett Usher that people in Rafah would have "nowhere to go" if Israeli troops launched their offensive.
"The safe areas that were declared are no longer safe. And if these people have to move - where can they move? We are really fearful of the horrific nature of where we are could only ever get worse," he said.
In other developments over the weekend:
- At least six Palestinians were killed in Israeli airstrikes on Rafah, according to Palestinian news agency Wafa
- On Saturday, the Israel Defense Forces (IDF) said its air force killed two Hamas operatives in the southern city
- The IDF also said it discovered a tunnel shaft near a school run by the relief agency for Palestinian refugees (UNRWA) that was leading to an "underground terrorist tunnel beneath UNRWA's main headquarters"
- UNRWA head Philippe Lazzarini denied any knowledge of a Hamas tunnel near the agency's office - a building which he said his staff vacated months ago
- A six-year-old girl who went missing in Gaza City last month was found dead with several of her relatives and two paramedics - after appearing to come under fire from Israeli tanks
- The IDF on Sunday said troops fighting in the southern city of Khan Younis had killed "approximately 100 terrorists"
- Three patients have died as Israeli troops prevented oxygen from getting to al-Amal Hospital in Khan Younis, the Palestine Red Crescent Society has said
The BBC is unable to independently verify many battlefield claims made during the course of the war.
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