The Israeli military said it had fired warning shots when crowds approached its forces.
Another 25 people, including several women and children, were killed in Israeli air strikes, according to local hospitals in Gaza. The military said it only targets Hamas militants.
The latest deaths came as Prime Minister Benjamin Netanyahu was expected to announce further military action — and possibly plans for Israel to fully reoccupy Gaza.
Experts say Israel's ongoing military offensive and blockade are already pushing the territory of some two million Palestinians into famine.
A new UN report said only 1.5 per cent of Gaza's cropland is accessible and undamaged.
Another escalation of the nearly 22-month war could put the lives of countless Palestinians and around 20 living Israeli hostages at risk, and would draw fierce opposition both internationally and within Israel.
Netanyahu's far-right coalition allies have long called for the war to be expanded, and for Israel to eventually take over Gaza, relocate much of its population and rebuild Jewish settlements there.
US President Donald Trump, asked by a reporter on Tuesday whether he supported the reoccupation of Gaza, said he wasn't aware of the "suggestion" but that "it's going to be pretty much up to Israel".
Of the 38 Palestinians killed while seeking aid, at least 28 died in the Morag Corridor, an Israeli military zone in southern Gaza where UN convoys have been repeatedly overwhelmed by looters and desperate crowds in recent days, and where witnesses say Israeli forces have repeatedly opened fire.
The Israeli military said troops fired warning shots as Palestinians advanced toward them, and that it was not aware of any casualties.
Nasser Hospital, which received the bodies, said another four people were killed in the Teina area, on a route leading to a site in southern Gaza run by the Israeli-backed Gaza Humanitarian Foundation, an American contractor.
The Al-Awda Hospital said it received the bodies of six people killed near a GHF site in central Gaza.
GHF said there were no violent incidents at or near its sites.
Two of the Israeli airstrikes hit Gaza City, in the north of the territory, killing 13 people there, including six children and five women, according to the Al-Ahli Hospital, which received the bodies.
The Israeli military says it only targets militants and blames civilian deaths on Hamas because its militants are entrenched in heavily populated areas.
UN experts say Israeli-backed aid group should be dismantled.
Israel facilitated the establishment of four GHF sites in May after blocking the entry of all food, medicine and other goods for 2.5 months.
Israeli and US officials said a new system was needed to prevent Hamas from siphoning off humanitarian aid.
The United Nations, which has delivered aid to hundreds of distribution points across Gaza throughout the war when conditions allow, has rejected the new system, saying it forces Palestinians to travel long distances and risk their lives for food, and that it allows Israel to control who gets aid, potentially using it to advance plans for further mass displacement.
The UN human rights office said last week that some 1400 Palestinians have been killed seeking aid since May, mostly near GHF sites but also along UN convoy routes where trucks have been overwhelmed by crowds. It says nearly all were killed by Israeli fire.
This week, a group of UN special rapporteurs and independent human rights experts called for the GHF to be disbanded, saying it is "an utterly disturbing example of how humanitarian relief can be exploited for covert military and geopolitical agendas in serious breach of international law".
