In effect, it would offer Israel recognition by the Arab world and the "normalisation" of relations in exchange for a full withdrawal from the territory Israel has occupied since the June 1967 Middle East war, including East Jerusalem.
Israel's Holocaust memorial and museum, in the forested outskirts of Jerusalem, is a wrenching reminder of the ashes from which the state arose, commemorating the 6 million Jews killed by the Nazis.
The White House said the visit was unofficial and private.
He also expressed confidence that a "renewed effort" could bring about peace between Israel and Palestine.
Prime Minister Netanyahu: Mr. President, it's a pleasure to see you again.
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He said: "We jointly held a belief that I would be entitled to the pension benefits arising from early retirement". The affidavits were filed in a high court application by the Democratic Alliance.
In another incident, Israeli leaders have tried to downplay any fallout from Trump's alleged revelation of sensitive Israeli information to the Russians, stressing that the ties between the intelligence communities will always be strong, without ever commenting on or confirming the report.
As Trump and Netanyahu continued to exchange greetings, the mood was electric at Rivlin's presidential residence in Jerusalem.
Trump promised to move the U.S. Embassy from Tel Aviv to Jerusalem while campaigning for president a year ago.
To say that Trump is going through the motions in terms of seeking a resumption of peace talks would be an exaggeration.
In Saudi Arabia, the president called on the Muslim world to combat radicalization. Wednesday he is scheduled to visit Pope Francis in Vatican City. Its 50th anniversary, according to the Jewish calendar, begins on Tuesday night, just hours after Trump leaves Israel. Since the 7th century C.E.it has been the site of Islam's Dome of the Rock and the Al-Aqsa Mosque. The chief rabbi of the Western Wall, Shmuel Rabinovich and his wife Yael, greeted them. Both men met separately with Trump at the White House in recent months. He placed a note between the massive stones, a centuries-old tradition of supplication.
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Roger Moore , the British actor best known for starring in James Bond films, died in Switzerland Tuesday at the age of 89, his family announced via Twitter.
Abbas, who also attended the Riyadh meeting, said that Trump had brought "broad horizons, prospects to recover our economy [and] continue building our nations based on the rule of law and non-violence ... building bridges instead of walls inside our land". A large temporary security screen blocked views of the Western Wall.
"This is a gesture for President Trump's visit, which does not harm Israel's interests", an Israeli government official told the Israeli media.
Despite his pro-Israeli campaign trail rhetoric, since entering the White House Mr Trump has caught some Israeli hard-liners off guard with the suggestion the government should "hold back" on settlement building, and his administration has equivocated over whether the United States embassy will move from Tel Aviv to Jerusalem as promised.
Turning to the prime minister, who joined him for the speech, Trump said, "Benjamin Netanyahu wants peace".
Trump said there were opportunities for cooperation across the Middle East: "That includes advancing prosperity, defeating the evils of terrorism and facing the threat of an Iranian regime that is threatening the region and causing so much violence and suffering".
Melania Trump visits a women's business center, school in Saudi Arabia
However, he warned that "the nations of the Middle East can not wait for American power to crush this enemy for them". Apart from Sunday's address, he's made no substantial remarks, other than exchanging pleasantries with other leaders.
"I have a feeling that we're going to get there eventually, I hope", Mr Trump said on Monday night, although few United States policy specifics have emerged to shed light on how the president plans to do so.