The president is "still looking at a lot of different facts", Kushner, who is also Trump's son-in-law, said during the keynote address at the annual Saban Forum in Washington on Sunday.
"When he makes his decision, he'll be the one who will want to tell you," he added.
Kushner's comments come just days before Trump is expected to make an announcement on the contentious issue.
Trump pledged during his campaign that he would move the embassy from Tel Aviv to Jerusalem, but has so far not fulfilled the promise.
Reports emerged on Friday that Trump could again delay moving the embassy, but recognise Jerusalem as the capital of Israel, a move that would rewrite long-standing US policy.
Many feel that moving the US embassy from Tel Aviv to Jerusalem would inflame tensions in the Middle East and upset the Palestinian leadership.
Israel declared Jerusalem to be its undivided and eternal capital in 1980, but the Israeli claim to the entire city remains unrecognised by the international community. Palestinians want occupied East Jerusalem as the capital of any future state.
No country currently has its embassy in Jerusalem - all such diplomatic missions are located in Tel Aviv.
Saeb Erekat, the secretary-general of the Palestine Liberation Organization, condemned this week any potential US recognition of Jerusalem as the Israeli capital.
Jerusalem is "the social, political, cultural, religious and economic heart of Palestine", he said in a statement.
"If the US administration decides to contradict its international commitments and historic foreign policy by recognising Jerusalem as the capital of Israel, it is not only going to promote international anarchy and disrespect for global institutions and law, but it will also be disqualifying itself to play any role in any initiative towards achieving a just and lasting peace," added Erekat.
PJ Crowley, the former US assistant secretary of state for public affairs, told Al Jazeera that "up until now, the US has deferred taking such a step because it was viewed as potentially prejudicial to final status negotiations between the parties".
"If Washington changes that posture and declares Jerusalem (or at least some of it) to be the Israel capital, then how will the Trump administration balance that gesture with the Palestinians," said Crowley, who is also the author of Red Line: American Foreign Policy in a Time of Fractured Politics and Failing States.
"We don't know, but it's not clear to me either why the White House feels that now is the right time to stir the hornet's nest."
At Sunday's Saban Forum, Kushner also spoke about US efforts to broker a peace deal between the Israelis and Palestinians.
He said he was hopeful because both sides "really trust" the US president.
"We do think it is achievable and we think that there is a lot of reasons why this is a time when it should happen," he said.
"The president sees [the conflict] as something that has to be solved," Kushner added.
Kushner said Israel needs to make peace with the Palestinians in order to normalise its relations with other Arab states in the Middle East.
"Israel, who is traditionally their foe, is a much more natural ally to them today than perhaps they were 20 years ago. Because of Iran, because of ISIS [the Islamic State of Iraq and Syria], because of extremism," Kushner said.
In a pre-taped video message shared earlier in the day, Benjamin Netanyahu, Israel's prime minister, echoed that sentiment, saying he believed Israel would become more openly accepted by its Arab neighbours in the coming years, "rather than in secret and the way it's done today".
Critics have repeatedly raised concern as to whether Kushner, and the Trump administration more generally, can be honest brokers between Israelis and Palestinians.
Late last year, US media widely reported that Kushner's parents had donated tens of thousands of dollars to Beit El, an Israeli settlement in the occupied West Bank, through a family-run foundation.
Kushner also reportedly served as the co-director of the Charles and Seryl Kushner Foundation from 2006 to 2015, something he failed to disclose in his financial records, according to Newsweek.
The Trump-appointed US ambassador to Israel, bankruptcy lawyer David Friedman, is also a staunch supporter of Israel's far-right and settler movements.