Since taking office, Trump has backed away from his campaign pledge to move the embassy to Jerusalem, saying now that he is still studying the issue.
In an unprecedented move on Monday, Israeli Prime Minister Benjamin Netanyahu published meeting notes from his visit with Trump at the White House in February, in order to affirm that he tried to convince Trump to relocate the U.S. embassy from Tel Aviv to Jerusalem.
However, the worldwide community does not recognize Israel's annexation of east Jerusalem, which the Palestinians seek as the capital of a future independent state.
Israel captured the Old City, along with the rest of east Jerusalem, in the 1967 Mideast war.
Israeli President Benjamin Netanyahu reportedly wanted to accompany Trump to the Western Wall, a request the White House reportedly denied.
"The ambassador explained why moving the embassy would help advance peace and not the opposite", the report released by the Prime Minister's Office said.
Separately, a reported comment by a USA official helping prepare Trump's visit also led to Israeli criticism, with the official allegedly telling Israeli counterparts that the Western Wall was part of the occupied West Bank.
Also on Tuesday, Trump's controversial choice for ambassador David Friedman presented his credentials in a ceremony at President Reuven Rivlin's residence in Jerusalem.
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Palestinians argue moving the embassy would prejudge one of the most sensitive issues in the conflict, undermining America's status as an effective mediator. It added: "Israel is certain that the comment contradicts President Trump's policy".
Since his inauguration, Trump has faced a flood of warnings from top Arab diplomats and a personal appeal from Jordan's King Abdullah to not move the USA embassy to Jerusalem from Tel Aviv.
Despite the passage of the Jerusalem Embassy Act of 1995 by the U.S. Congress, which required the relocation of U.S. Embassy in Israel from Tel Aviv to Jerusalem, former U.S. presidents, including George W. Bush, Bill Clinton and Barack Obama, consistently renewed a presidential waiver to delay the relocation on national security interests.
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He has also advocated moving the embassy to Jerusalem. Israel began its occupation of the West Bank at the same time; the occupation is regarded as illegal by most countries, though Israel disputes that.
Netanyahu told Friedman his immediate visit to the Western Wall was "a strong gesture of solidarity".
Abbas is also at risk of being sidelined by Trump's efforts to revive Israeli-Palestinian peace talks. "We want to see our country established on our sovereign territories with East Jerusalem as our state's capital", Abu Rudeineh said.