But concerns over a funding cut straddle both sides of the conflict. The United States contributes roughly $700 million annually to the Palestinians, about half of it through the United Nations agency charged with humanitarian relief for Palestinian refugees, UNRWA.
"Cutting funding would not bring anything good to the situation," said an Israeli security official, speaking on the condition of anonymity because of the sensitivity of the topic. "Doing this would end up making the Palestinian leadership even weaker."
From the outset, Trump's quest to secure what he has described as the ultimate deal by brokering a Middle East peace process has appeared to the Palestinian side to be inherently skewed, given that it is led by Trump's son-in-law and adviser Jared Kushner, whose family supports settlements that the Palestinians see as illegally built in their future state.
The dealbreaker, though, came with Trump's decision last month to recognize Jerusalem as Israel's capital, which had Palestinian President Mahmoud Abbas walking away before a plan had even been presented, saying it ignored Palestinian claims to the city.
Representatives of the main Palestinian factions, including the militant group Hamas and Islamic Jihad, are set to meet later this month to decide on a response to the latest U.S. moves. Abbas, who has rejected armed conflict with Israel and whose security forces cooperate with the Israeli military and share intelligence, is already struggling politically. He is now likely to come under pressure to take a stronger stance or see himself further weakened.
Other Palestinian factions have criticized Abbas's cooperation with Israel as collusion with the enemy.
If his Palestinian Authority, which has long benefited from U.S. funding, is weakened to a level where it collapses, "then there really would be no one to talk to or rely upon," the Israeli security official said.
The tweet Trump sent Tuesday evening threatening to cut aid was particularly inflammatory to Palestinians because it also said that Jerusalem had been taken "off the table" for negotiations, although the wording wasn't clear.
The White House did not attempt to clarify Trump's tweets Wednesday morning, leaving uncertainty as to whether the president was walking away from his administration's earlier assurances that the U.S. policy shift on Jerusalem would not preclude Palestinian claims to East Jerusalem in future negotiations.
Later, White House press secretary Sarah Huckabee Sanders was asked whether Trump is signaling a larger change in the U.S. position on borders and negotiations.
"I don't think so," she replied. "It doesn't affect the negotiations. We still want to continue to have conversations and continue the peace process. We're still very much committed to that and hope we can continue to push forward in that point."
A U.S. official said there had been no word of such a shift either on the final status of Jerusalem or, in the days leading up to the tweets, about aid to the Palestinians. It was also not clear whether Trump was threatening to cut off all aid to the Palestinians.
"He doesn't want to give any additional funding until the Palestinians agree to come back to the negotiation table," Haley said at the United Nations on Tuesday in response to a question on funding for UNRWA, the U.N. Relief and Works Agency, which receives 40 percent of its annual budget from the United States. "We're trying to move for a peace process, but if that doesn't happen, the president is not going to continue to fund that situation."
Other U.S. funding for the Palestinian Authority was already threatened by legislation moving through Congress that would cut off aid unless Palestinians can show it is not going to terrorists or their families.
Widespread protests against Trump's plan to move the U.S. Embassy to Jerusalem in which 12 Palestinians were killed have largely died down. But Palestinian militants have fired rockets from Gaza at a rate not seen since the 2014 war between Israel and Hamas, which controls the Palestinian enclave.
Three more rockets were fired toward Israeli territory Wednesday, the Israeli military said.
It is the impoverished Gaza Strip, where UNRWA operates schools, health clinics and other community projects, that is likely to be most affected by any cut to Palestinian funding.
Some Israeli political leaders have lobbied for an end to U.S. funding to UNRWA, saying that its schools were used by Hamas during the latest conflict with Israel.
But some of Israel's defense officials see the agency's work as an important counterbalance to Hamas, designated as a terrorist entity by Israel, the United States and the European Union.
"Traditionally, the Israeli defense establishment has resisted pressure by Israeli hawks who want to shut down UNRWA funding," said Ofer Zalzberg, a senior analyst at the International Crisis Group. "They say, if it's not UNRWA, then education will be provided by Hamas."
The U.N. agency also works in refugee camps in the West Bank and Gaza where daily life for Palestinians is bleakest and armed factions find fertile ground for recruitment.
Chris Gunness, UNRWA's spokesman, said that the agency had not been notified of any changes to U.S. funding.
"Cutting aid to the Palestinians at this stage would have the opposite effect to what the Americans want," said Moshe Maoz, an Israeli professor of Islamic and Middle Eastern studies at the Hebrew University of Jerusalem. "This is brutal pressure from the U.S. The situation in Gaza is terrible. If America cuts its aid, it would be catastrophic."
In an article published in November on the website of the Institute for National Security Studies, Maj. Gen. Yoav Mordechai, head of COGAT, the Israeli military authority responsible for implementing government policy in the West Bank, urged the international community to increase funding to Gaza. Despite tightly restricting trade and movement, there is concern about the security fallout from a worsening humanitarian crisis.
Alon Eviatar, a former adviser on Palestinian affairs to COGAT, said the fact the United States is even considering cutting funding to UNRWA is a significant change. In the past, he said, the agency was always treated as holy.
"If the U.S. pulled out of its commitment to UNRWA, it could have a significant impact and even a domino effect on other countries funding it," he said.
UNRWA has been fighting to save its budgets for much of the past decade as other world issues have become more prominent and the Palestinian refugee issue seen as less of a problem.
The United States pays "the Palestinians HUNDRED OF MILLIONS OF DOLLARS a year . . . with the Palestinians no longer willing to talk peace, why should we make any of these massive future payments to them?" Trump tweeted Tuesday.
Much of the remaining funding provided by the United States is distributed through USAID projects.
"The economic situation of the people of Gaza is very important from a security perspective," said retired Gen. Eli Ben-Meir, former intelligence chief for the Israeli military. He added, however, that in his opinion UNRWA wasn't the best organization to distribute aid.
Haley said Tuesday that U.S. officials "still very much want to have a peace process," but it quickly appeared that the strong-arm tactics could drive Palestinians further from the negotiating table.
"It's being perceived as deeply offensive," Zalzberg said of Trump's threat. "It's been taken to say, 'We will pay you to make a concession on Jerusalem.' "
Palestinian officials say they are willing to return to negotiations, but not under coercion.
"Jerusalem and its holy sites are not for sale, not with gold, nor with silver," Nabil Abu Rudeineh, a spokesman for Abbas, said in a statement. He said the Palestinian negotiations need to have international legitimacy and be based on a two-state solution.
Still, Trump's tweets were welcomed by some right-wing Israelis. Education Minister Naftali Bennett, leader of the ultranationalist Jewish Home party, commended the president for not being afraid "to speak the truth, even if it is not popular."
"The truth is no peace deal could ever be predicated on the division of Jerusalem," he said in a statement."The truth is the Palestinian leadership continues to fund terrorists, using US tax monies. By speaking truth, we can stop the fantasy of another Palestinian state in the heart of Israel."
Sufian Taha in Jerusalem and Anne Gearan in Washington contributed to this report.