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.
Speaking to Meet the Press, Secretary Tillerson stated "The president is being very careful to understand how such a decision would impact the peace process".
Earlier on Monday Conor Powell, a Jerusalem-based reporter for the USA network, tweeted: "Everyone I've spoken to in DC that has been briefed on Jerusalem embassy move says Netanyahu told Trump not to move embassy at this time".
Israel occupied the West Bank and east Jerusalem in 1967. "It will advance the process because of the correction of the historic injustice and the smashing of the Palestinian fantasy of Jerusalem not being the capital of Israel", it said.
It was not the first time a member of the Trump administration referred to "Palestine", which the US does not recognize. In recent weeks, press reports in Israel indicated that the Trump administration was not planning to move the embassy.
In a move unrecognized pretty much by the rest of the world, Israel declared Jerusalem al-Quds its capital in 1980s.
"This is Prime Minister Netanyahu's consistent and clear stance", his office clarified.
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Most chapters are clearly outdated and do not reflect the most recent standards in USA trade agreements". Kevin Brady (R., Texas), chairman of the House Ways and Means Committee, which oversees trade.
The Israeli officials were reportedly told that the President's visit to the site was private and that he did not want Prime Minister Benjamin Netanyahu to accompany him there; the American official reportedly added that Israel did not have jurisdiction over the area.
President Trump is the latest in a long line of United States presidents who have tackled the Israel-Palestine peace process early in their first term, and while most quickly lost interest, Trump vowed to get the peace deal done with "whatever is necessary".
The US president will start his first ever foreign tour by visiting the Saudi capital on Friday and then will leave for Israel on Monday.
Trump is to begin his first trip overseas as president on May 19, visiting Saudi Arabia, Israel (and the West Bank city of Bethlehem), and the Vatican before jetting to a North Atlantic Treaty Organisation meeting in Brussels and the G7 summit in Sicily. He is scheduled to meet Netanyahu in Jerusalem before travelling to Bethlehem to meet with Palestinian President Mahmoud Abbas.
The US and other countries have their embassies in Tel Aviv.
The former bankruptcy lawyer has expressed scepticism over the two-state solution to the Israeli-Palestinian conflict, the basis of years of USA peace efforts.
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.