The American and Israeli teams were visiting the holy site on Sunday ahead of Trump's visit next week.
The Israeli team requested to have Prime Minister Benjamin Netanyahu accompany Trump during his visit to the Western Wall, but were turned down by the Americans. This led to a loud argument between representatives from the two teams.
Two officials from the US Consulate in Jerusalem spoke curtly to the Israeli team, telling them among other things: "We're asking you to leave, and we need to be left alone. Israeli officials cannot be here; this is not your territory, it's the West Bank. This is a private visit by the President and has nothing to do with you. It's none of your business."
The shocked members of the Israeli team told the Americans, "This is completely unacceptable, and we reject these statements out of hand. This is a holy place under Israeli sovereignty."
The American delegation then insisted on separating from the Israeli team and continued their visit to the Western Wall and the Church of the Holy Sepulchre on their own.
The American team also rejected Israeli suggestion to have Government Press Office teams document President Trump's visit to the holy sites in Jerusalem and broadcast it live. "This is a private visit," the Americans insisted.
The Israeli team reported the American comments to the Prime Minister's Office.
A PMO official said in response, "The claim that the Western Wall is in the West Bank was met with shock. Jerusalem is convinced this statement is at odds with President Trump's policy, as it was expressed in the firm American objection to the latest UN Security Council resolution. Israel is seeking clarifications from the US on this matter."
The new US Ambassador to Israel David Friedman arrived at Ben-Gurion Airport on Monday afternoon and headed straight for the Western Wall to pray, where he had a chance encounter with Aerosmith frontman Steven Tyler.