The Palestinian news agency Wafa did not say when the letter was sent but said it aimed to explain the "risks" of moving the embassy from Tel Aviv to Jerusalem.
Abbas warned the move would have a "disastrous impact on the peace process, on the two-state solution and on the stability and security of the entire region," Wafa said.
Trump has said he plans to relocate the embassy from Tel Aviv to Jerusalem in a controversial move bitterly opposed by Palestinians as a unilateral action while the status of the city remains contested.
Abbas also sent letters to other world powers including Russia, China and the European Union, calling on them to "spare no effort" to prevent the United States from making the move, Wafa said.
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The Palestinians regard Israeli-annexed east Jerusalem as the capital of their future state, while Israel proclaims the entire city as its undivided capital.
The United States and most UN member states do not recognise Jerusalem as Israel's capital, and the city's status is one of the thorniest issues of the decades-long Israeli-Palestinian conflict.
Trump has also nominated David Friedman, a supporter of Israeli settlements in the West Bank, as his ambassador to the Jewish state.
In a Trump transition team statement last month, Friedman said he wanted to work for peace and looked forward to "doing this from the US embassy in Israel's eternal capital, Jerusalem".
Israel captured east Jerusalem in the Six-Day War of 1967 and subsequently annexed it in a move never recognised by the international community.