Canberra "will carefully examine the arguments put forward by Australia's former Ambassador to Israel, Dave Sharma, that we should consider recognising Jerusalem as the capital of Israel, without prejudice to its final boundaries, while acknowledging East Jerusalem as the expected capital of a future Palestinian state", the joint statement reads.
He added that, in the next few months, he would "confer with cabinet colleagues" and "with other leaders around the world to gauge the perception of this and to make the case".
Israeli Prime Minister Benjamin Netanyahu said he had recently spoken to Morrison and welcomed the Australian policy shift.
Palestine's Maliki, who is in Jakarta as part of a week-long celebration of Palestinian solidarity by Indonesia, said Australia was at risk of violating global law, as well as its business relations with the rest of the world.
Trump's move ruptured decades of worldwide consensus that Jerusalem's status should be settled as part of a two-state peace deal between Israel and the Palestinians.
The decision prompted the 13 ambassadors to call a meeting in the Australian capital, Mohamed Khairat, Egypt's ambassador to Australia told Reuters by telephone.
Mr Morrison has also announced the government would review Australia's support for the Iran nuclear deal to see whether it was still "fit for purpose" in preventing the country from acquiring nuclear weapons after Mr Trump withdrew the U.S. from the Obama-era deal.
"What I am doing today is recognising what is a real concern in the Jewish community in Australia - whether it is about the Iran nuclear deal, which is an item of great concern to that community, or it is on this broader question of the two state solution and how with go about achieving it. We will continue to strengthen ties between", he said in the Tweet.
He described proposals to recognise Jerusalem and move Australia's embassy as "sensible" and "persuasive" and would be considered by the government.
No comment was immediately available from Morrison's office.
Labor accused Mr Morrison of "desperation", with Shadow Foreign Affairs Minister Penny Wong saying the Prime Minister was playing "dangerous and deceitful" word games.
The status of Jerusalem is one of the thorniest obstacles to a peace deal between Israel and the Palestinians. In their joint statement, Morrison and Payne said they would oppose a resolution in the United Nations General Assembly that would allow the "State of Palestine" to take over the chairmanship of the Group of 77, an important regional bloc at the UN.
Israel regards all of the city, including the eastern sector that it illegally annexed after the 1967 Middle East war, as its capital while the Palestinians, with broad global backing, want East Jerusalem as the capital of a future state that they hope to establish in the occupied West Bank and the Gaza Strip.