Israel's right-wing had placed high hopes in Trump's presidency following his pledges of ardent support for Israel.
Separately, a reported comment by a US 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.
The Western Wall, the holiest site where Jews can pray, is located in east Jerusalem, which was occupied by Israel in 1967 and later annexed in a move never recognised by the international community.
Palestinians see east Jerusalem as the capital of their future state.
Both highly sensitive issues made headlines in Israel on Tuesday as preparations intensified for Trump's visit to the country and the Palestinian territories on May 22 and 23.
Trump's controversial new ambassador also presented his credentials to the Israeli president at a ceremony in Jerusalem.
Israel's right-wing had placed high hopes in Trump's presidency following his pledges of ardent support for Israel and a commitment to move the embassy from Tel Aviv to the disputed city of Jerusalem.
Some even called for the end of the idea of a Palestinian state.
But Trump has since backed away from the embassy move, saying it was still being looked at, while Israeli Prime Minister Benjamin Netanyahu has faced political backlash at home over the issue.
A Fox News reporter cited sources saying Netanyahu had asked Trump not to move the embassy now, leading the prime minister to issue an angry rebuttal.
In an unusual move, Netanyahu also partially released the minutes of a meeting he had with Trump in Washington in February, showing he had pressed the new president to move the embassy.
The rival claims to Jerusalem lie at the heart of the Israeli-Palestinian conflict and no countries currently have their embassies in Jerusalem, instead basing them in Israel's commercial capital Tel Aviv.
Palestinians say moving the embassy would be a de facto recognition of Israel's control of all Jerusalem.
Separately, Israeli officials have reacted angrily to reports in local media that representatives of the American consulate had suggested the Western Wall did not belong to Israel.
Trump is reportedly planning to visit the wall in Jerusalem's Old City and Israel's Channel Two reported that Israeli officials offering to help plan the event were told by American counterparts it was not their remit.
One US official said the Western Wall was part of the West Bank, Channel Two reported.
The White House distanced itself from the alleged comments, saying in statements to US media that they did not reflect the views of the adminstration.
The US, like most of the international community, considers Jerusalem disputed ahead of final-status negotiations.
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.
Jewish-American Friedman, a strong supporter of Israeli settlement building in the West Bank, entered the residence with a marching band playing.
Speaking after the ceremony, Friedman did not directly respond to the dispute over the Western Wall but pledged to "support the state of Israel in every way".
"I pledge to you to do all that I can to strengthen and enhance the relationship between our two great nations," he told Rivlin.
Speaking of Trump, he said: "His love for and commitment to the state of Israel is rock solid and it enjoys his highest priority."
Rivlin called on the "whole world to recognise Jerusalem as the official capital of the state of Israel".
Friedman arrived Monday and immediately visited the Western Wall, praying there and kissing the sacred site.
The former bankruptcy lawyer has expressed scepticism over the two-state solution to the Israeli-Palestinian conflict, the basis of years of US peace efforts.
He has also advocated moving the embassy to Jerusalem.