Wafa, the official news agency of the Palestinian Authority (PA), reported on Tuesday that Trump called Abbas to inform him about his "intention".
"President Abbas warned of the dangerous consequences such a decision would have to the peace process and to the peace, security and stability of the region and of the world," Nabil Abu Rudeina, the Palestinian president's spokesperson, said. No further details were given about when Trump plans to move the embassy.
Jerusalem's status is an extremely sensitive aspect of the Israeli-Palestinian conflict. Israel claims the city as its capital, following the occupation of East Jerusalem in the 1967 war with Syria, Egypt and Jordan, and considers Jerusalem to be a "united" city.
Palestinians have long seen East Jerusalem as the capital of their future state.
No country currently has its embassy in Jerusalem, and the international community, including the US, does not recognise Israel's jurisdiction over and ownership of the city.
In recent days, senior Palestinian officials warned of the potentially destructive effects of any move denying their claim to occupied East Jerusalem as the capital of their future state.
Saeb Erekat, the secretary-general of the Palestine Liberation Organization, said this week he had spoken to Arab leaders, who relayed the message "that Jerusalem is a red line, not just for Palestinians but for Arabs, Muslims and Christians everywhere".
During his election campaign last year, Trump repeatedly promised to move the embassy and recognise Jerusalem as the Israeli capital.
In June, however, like his predecessors, Trump signed a six-month waiver to delay the relocation, which would have complicated US efforts to resume the long-stalled Israeli-Palestinian peace talks.
"Every US president for more than 20 years has recognised that this could have cataclysmic results and repercussions if it were to happen," Al Jazeera's Kimberly Halkett, reporting from Washington, DC, said.
A number of world leaders have sharply criticised such a relocation, fearing it would further escalate regional tensions.
Turkish President Recep Tayyip Erdogan on Tuesday threatened to cut diplomatic ties with Israel over reports that the US plans to recognise Jerusalem as the Israeli capital.
Such a move would be a "red line" for Muslims, he said.
French President Emmanuel Macron told Trump by telephone that Jerusalem's status must be decided in peace talks between Israelis and Palestinians.
Also on Tuesday, the Arab League held an emergency meeting to discuss developments on the status of Jerusalem, following a request by Palestinian officials.