The alleged incident took place at a site run by the Gaza Humanitarian Foundation (GHF), which is backed by Israel and the US but condemned by much of the rest of the international community.
A former security contractor at one of the sites told the BBC he saw a guard with a machine gun open fire from a watchtower because a group of women, children and elderly people were moving away from the site too slowly.
Another contractor then opened fire, the whistleblower claimed. He said: “A Palestinian man dropped to the ground motionless. And then the other contractor, who was standing there, was like ‘damn, I think you got one’. And then they laughed about it.”
The Associated Press reportedly spoke to two contractors for UG Solutions, who have been sub-contracted to GHF sites, and alleged that there was regular use of live ammunition, stun grenades and pepper spray on Palestinians who posed no threat.
They claimed security staff hired to protect the sites were often unqualified and unvetted, and acted with impunity while heavily armed.
The GHF said the accusations were categorically false, adding that no civilians had ever come under fire at its distribution sites.
The new model for aid delivery has been designed to prevent food and supplies falling into the hands of Hamas, thereby propping up the terror group.
Rather than delivering aid into population centres, the system requires people to walk long distances across open ground towards specially created centres.
Photos and footage have shown large numbers of people being forced into narrow corridors between berms of earth, with security contractors standing above them.
Hundreds have been killed in shootings in the wider vicinity of the aid centres as eyewitnesses have accused Israeli troops, who provide a wider ring of security, of opening fire. Israel has denied this, but has admitted firing warning shots in the area.
However, the new testimony has marked the first time an American staff member at the sites has been implicated.
The GHF said it was operating in the face of people with a “vested interest” in seeing it fail and denied any misconduct or lack of experience among staff.
The organisation said it had distributed the equivalent of more than 50 million meals in Gaza.