The Gaza Humanitarian Foundation said 20 people died at an aid point in Khan Yunis, in the south of the Palestinian territory on Wednesday, blaming armed "agitators" in the crowd.
"Our current understanding is that 19 of the victims were trampled and one was stabbed amid a chaotic and dangerous surge," the US- and Israel-backed organisation said in a statement.
A medical source at Nasser Hospital in Khan Yunis gave a lower toll, telling AFP it had received "nine martyrs, including several children" after "Israeli forces" opened fire.
The deaths came as Gaza's civil defence agency said eight people were killed in Israeli strikes across the Palestinian territory on Wednesday, including six in Gaza City.
There was no immediate comment from the Israeli military when contacted by AFP.
The GHF said the stampede was "driven by agitators in the crowd", adding: "We have credible reason to believe that elements within the crowd -- armed and affiliated with Hamas -- deliberately fomented the unrest".
The medical source at Nasser Hospital said the victims were "heading to the aid distribution centre in northwest Rafah to receive food aid" but the main gate to the centre had been closed.
"The Israeli occupation forces and the centre's private security personnel opened fire on them, resulting in a large number of deaths and injuries," the source added.
The GHF, an officially private effort, began operations on May 26 after Israel had halted supplies into the Gaza Strip for more than two months, sparking warnings of imminent famine.
On Tuesday, the UN said it had recorded 875 people killed in Gaza while trying to get food, including 674 "in the vicinity of GHF sites", since late May.
Last week, UN rights office spokeswoman Ravina Shamdasani told reporters that "most of the injuries are gunshot injuries".
The GHF has denied that fatal shootings have occurred in the immediate vicinity of its aid points and the Israeli army has accused Hamas of being responsible for firing at civilians.
"For the first time since operations began, GHF personnel identified multiple firearms in the crowd, one of which was confiscated," the GHF said on Wednesday.
"An American worker was also threatened with a firearm by a member of the crowd during the incident," it added, calling it part of a "deeply troubling pattern", including "false messages" about aid site openings.
With AFP
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