The new US president campaigned for office on a long-standing Washington plan to move its main mission to Israel to the city that remains the focus of its conflict with the Palestinians.
To have done so would have been read by many as a signal that Washington supports Israel's claim on Jerusalem as the eternal and undivided capital of the Jewish state.
This would reassure Israel's internationally isolated right-wing government but trigger fury among the Palestinians, still hoping to make east Jerusalem the capital of a future state.
And it would have complicated Israel's delicate efforts to build a network of open and covert alliances with the broader Arab world as a bulwark against Iran and Islamist extremism.
The question is bound to come up on Wednesday, when Trump welcomes Israel's Prime Minister Benjamin Netanyahu to the White House and sets the tone for future US-Israeli ties.
In 1995, the US Congress passed a law ordering that the US mission be shifted from Tel Aviv to the Holy City, but every president since has delayed taking a decision on the move.
In the meantime, the debate has become a proxy for that over US support for Israel's settlement building enterprise on occupied Palestinian land in the West Bank and east Jerusalem.
The previous US administration under president Barack Obama opposed the expansion of Jewish settlements, arguing that they hurt the longer-term search for a two-state solution.
But this did not hold back Trump the presidential candidate, who promised to move the embassy "fairly quickly" if he were to win.
Trump the president has been less determined.
"I am not somebody that believes that going forward with these settlements is a good thing for peace," Trump reportedly told the Hebrew newspaper Israel Hayom last week.
Moving the embassy, he admitted, was "not an easy decision."
This would seem to fly in the face of the view of David Friedman, the lawyer tipped to become US ambassador to Israel.
In a news release to mark Trump's decision to nominate him, Friedman said he looked forward to working from "the US embassy in Israel's eternal capital, Jerusalem."
But observers in Washington agree that Trump appears to have moved away from a quick decision, towards the more traditional wait-and-see approach adopted by his recent predecessors.
"The Israelis are obviously in favor of the move," said Jonathan Schanzer, vice president of the Foundation for Defense of Democracies, a conservative think tank.
"But I got the sense from Israeli officials on a visit in January that the Israelis are willing to tread slowly."
Schanzer said his impression was that Israel would not push on the embassy issue if it is assured that Washington remains loyal to its primary security objectives in the region.
Robert Satloff of the Washington Institute of Near East Policy said Israelis across the political spectrum would welcome a move of the embassy but that the timing is crucial.
This year marks the 50th anniversary of the Six Day War -- when in Israeli eyes their victory secured the "reunification" of Jerusalem -- a sensitive date for Palestinians.
Under the Hebrew calendar, the commemoration falls in May. Coincidentally, this is also when Obama's last six-month "waiver" delaying the move expires.
"If the president wants to move forward on this, he should do so well in advance of that date lest he inadvertently feed any Palestinian sense of provocation and outrage," Satloff said.
Sooner than May does not seem likely, however.
At his first news conference after last month's inauguration, White House spokesman Sean Spicer said: "We are at the very beginning stages of even discussing this subject."
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