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Israel has long viewed the UN agency, known as UNRWA, as biased against it, an allegation the agency strongly denies, saying it is only providing necessary services to Palestinians.
Israeli officials also criticise the agency's method of classifying refugees, with descendants also eligible to register.
He said that while millions of other refugees around the world were cared for by the office of the UN High Commissioner for Refugees (UNHCR), the Palestinians have their own body which also treats "great-grandchildren of refugees -- who aren't refugees."
"This absurd situation must be ended," Netanyahu said.
In June, Netanyahu said he had raised the issue with Washington's UN envoy Nikki Haley.
On Wednesday, Trump threatened to cut aid worth more than USD 300 million annually to the Palestinians in a bid to force them to negotiate.
The United States has long provided the Palestinian Authority with much-needed budgetary support and security assistance, as well as an additional USD 304 million for UN programmes in the West Bank and Gaza.
A Friday report on Israeli Channel 10 television said the US had frozen a payment due to UNRWA, but a spokesman for the UN organisation said yesterday that they "have not been informed directly of a formal decision either way by the US administration."
UNRWA runs hundreds of schools for Palestinian refugees in the Gaza Strip, the occupied West Bank, Israeli-annexed east Jerusalem, Lebanon, Jordan and Syria.
It also distributes aid and provides teacher training centres, health clinics and social services.
Many analysts, including Israelis, warn that closing the agency without having an effective replacement could lead to further poverty and perhaps violence.
"While UNRWA is far from perfect, the Israeli defence establishment, and the Israeli government as a whole, have over the years come to the understanding that all the alternatives are worse for Israel," Peter Lerner, a former spokesman for the Israeli military, wrote in an opinion piece in Haaretz newspaper last week.
"In an extreme situation, the administration of those refugees could fall on Israel's shoulders.