The decision disregarded normal procedures and will add to Palestinian fury and global concern over the diplomatic relocation from Tel Aviv.
Several Israeli media reports said the longtime premier was questioned at his official Jerusalem residence over a case of alleged corruption involving local telecoms giant Bezeq. The move is timed to coincide with the 70th anniversary of the State of Israel's founding.
The waiver will be valid for three years.
Trump has heralded his December 6, 2017 recognition of Jerusalem as Israel's capital and the move of the embassy as a highlight of his administration.
"We will not allow bureaucracy needlessly to delay the transfer", he said. The third phase, the site selection and construction of a new embassy, will take up to nine years.More news: Investigators search Cambridge Analytica's London offices
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The new embassy initially will be housed in southern Jerusalem, in the Arnona neighborhood, on a compound that now houses the consular operations of the Consulate General of Jerusalem.
Until now, the United States embassy has been located in Tel Aviv with a separate consulate general located in Jerusalem that represents U.S. interests in the Palestinian territories.
"I am looking at coming, and if I can, I will", Trump said.
In his speech, Mr Abbas criticised the USA recognition of Jerusalem as Israel's capital, the American plan to move its embassy to the city and the cut-off of hundreds of millions of dollars in aid to the United Nations agency for Palestinian refugees.
Palestinian families are planning to pitch hundreds of tents in the Gaza Strip near the border starting on Friday, known as "Land Day", which marks the Israeli government's expropriation of Arab-owned land in the Galil on March 30, 1976, and the ensuing demonstrations in which six Arab Israelis were killed.