The humanitarian charity said displaced people were injured "due to intense gunfire from the Israeli drones targeting citizens at Al-Amal Hospital" on Friday, as well as the rescue agency's base.
The Israeli military said it was checking the report, which comes amid a major advance in Khan Younis.
Nearby Al-Amal, Israeli tanks were also reportedly approaching Gaza's biggest remaining functioning hospital, Nasser, where people reported hearing shellfire from the west. Residents also reported fierce gun battles to the south.
Israel has launched a major new advance in Khan Younis this week to capture the city, which it says is now the primary base of the Hamas fighters who attacked Israeli towns on October 7, precipitating a war that has since devastated the Gaza Strip.
Gaza's Hamas-run health ministry said on Friday 142 Palestinians had been killed and 278 injured in the enclave in the past 24 hours, taking the death toll from more than three months of war there to 24,762.
The World Health Organization says most of the enclave's 36 hospitals have stopped working.
Only 15 are partially functioning, and those are operating at up to three times their capacity, without adequate fuel or medical supplies, it says.
Israeli officials have accused Hamas fighters of operating from hospitals, including Nasser, which staff deny.
More than 1.7 million people - around 75 per cent of Gaza's population - are estimated to be displaced, many forced to move repeatedly, according to UN Palestinian refugee agency (UNRWA) figures.
Many have sought refuge in tents that do little to protect them from the elements and disease.
While saying he was not shying away from the "human tragedy" inflicted on Gaza civilians, Israeli President Isaac Herzog cast the offensive as a step towards more peaceful relations with the Palestinians in the future, and bolstering global security, during his appearance at the World Economic Forum in Davos.
In the north, where Israel says it has started pulling out troops and shifting to smaller scale operations, 12 people were killed in Israeli strikes on a residential building near the largely non-functioning Al Shifa Hospital in Gaza City, Palestinian health officials said.
Israel's onslaught on Gaza was triggered by Hamas attacks in which around 1,200 people were killed and 253 taken hostage, of whom about half are still in Gaza, according to Israeli tallies.
Israel says it will fight on until Hamas is eradicated, an aim Palestinians call unachievable because of the group's structure and deep roots in an enclave it has run since 2007.
Diplomats were dealing on Friday with the repercussions after Israeli Prime Minister Benjamin Netanyahu appeared to rule out an independent Palestinian state, rejecting a long-standing pillar of US strategy in the Middle East.
"Israel must have security control over the entire territory west of the Jordan River," Mr Netanyahu told a briefing in Tel Aviv on Thursday. "It clashes with the principle of sovereignty, but what can you do?"
US State Department spokesperson Matthew Miller responded at a news briefing that the establishment of a Palestinian state was the only way to provide lasting security to Israel itself, along with reconstruction, governance and security for Gaza.
The UK Government on Friday slammed Mr Netanyahu for his "disappointing" rejection of any Palestinian state.
AFo Office spokesman told the Standard: "Prime Minister Netanyahu's comments on a future Palestinian state are disappointing."The UK's position is very clear. A two-state solution, with a viable and sovereign Palestinian state living alongside a safe and secure Israel, is the best route to lasting peace."