He has approved a waiver to defer relocating the embassy, as his predecessors have done every six months since 1995.
Palestinians claim the city's east and have warned against a United States move.
Israeli Prime Minister Benjamin Netanyahu's office said in a statement that Israel was "disappointed" by Trump's decision, arguing that the presence of embassies outside the city "drives peace further away by helping keep alive the Palestinian fantasy that the Jewish people and the Jewish state have no connection to Jerusalem".
The Palestinians are praising President Donald Trump's decision not to move the U.S. Embassy to Jerusalem, saying it strengthens the chances of peace.
The same official said that while Trump may not authorize the move this week, the President still envisions moving the embassy at some point.
"As he has repeatedly stated his intention to move the embassy, the question is not if that move happens, but only when", the statement from the White House read.
2 children believed to have been shot by mother die
Police say there are indications the mother has mental health issues but it hasn't been determined why the children were shot. Online jail records show Helton was held on two counts of attempted aggravated murder and four counts of felonious assault.
Presidents of both parties have consistently renewed the waivers for six-month stretches.
"As much as he wanted the move to occur", former U.S. Ambassador Dan Shapiro wrote on Twitter, "he didn't want it forced when it could disrupt the overriding strategic goal of completing successful negotiations [with] the Palestinians".
"We are ready to start the consultation process with the USA administration", he said after Trump's announcement.
Palestinian leaders cheered the move and said it improved the atmosphere for future negotiations by demonstrating Trump's seriousness about the process. Israel captured east Jerusalem - claimed by Palestinians for the capital of a future independent state - from Jordan in 1967 and annexed it, in a move not internationally recognized. Both are in the Old City, part of east Jerusalem.
His office said it believes all embassies should be based in what it called Israel's "eternal capital".
A senior Israeli official is expressing disappointment over Trump's decision against relocating the embassy to Jerusalem and is accusing the US of caving in to Arab pressure. The US president has said he is reviewing whether to fulfil his campaign promise to move the facility to Jerusalem. Trump avoided any public mention of a potential embassy move during his visit to Israel and the West Bank in May. The long-standing USA position is that Jerusalem's fate must be worked out through Israeli-Palestinian negotiations.
Abdullah in particular expressed deep concern that moving the U.S. Embassy would spark unrest among his country's large Palestinian population.