Moon's trip comes after his visit last week to the communist North where he held his third bilateral summit with leader Kim Jong-un.
Moon will explain the outcome of the latest inter-Korean summit before the entire world when he addresses the United Nations General Assembly. However, his U.S. trip may be more aimed at explaining the summit outcome directly to the U.S. president.
"President Moon is set to thoroughly explain the outcome of the third inter-Korean summit (in 2018) that was held under the interest of the entire world," said Nam Gwan-pyo, a senior director from the presidential National Security Office.
Moon "will hold in-depth discussions (with Trump) on ways to break the impasse in North Korea-U.S. talks and to improve North Korea-U.S. relations," Nam told reporters earlier.
The Moon-Trump summit is scheduled to be held Monday (New York time).
The latest Moon-Kim summit marked the third of its kind held since Moon took office in May 2017 and was the fifth inter-Korean summit in history.
The South Korean president's trip to Pyongyang, from Tuesday to Thursday, came amid a deadlock in denuclearization talks between the U.S. and the North, partly caused by Trump calling off a scheduled North Korea trip by his Secretary of State Mike Pompeo, citing a lack of progress in the North's denuclearization process.
In a joint declaration with Moon, Kim agreed to dismantle the North's missile engine test facility and launch pads in Dongchang-ri, which is considered home of the North's newly developed long-range ballistic missiles that can reach the continental U.S.
The North has also offered to dismantle its key nuclear facilities in Yongbyon in exchange for corresponding measures from the U.S.
Moon said he had additional messages for Trump that come directly from the North Korean leader.
"Among what we discussed, there are items that we did not include in the joint declaration," Moon told a nationally televised press conference after his return from North Korea on Thursday. "I plan to deliver such messages in detail to the U.S. side."
Moon said the North Korean leader repeatedly confirmed his commitment to denuclearize as quickly as possible.
Kim wants to hold a second North Korea-U.S. summit to do so, Moon told the press conference.
"I believe the denuclearization process may move forward much faster should the two leaders sit face to face," the South Korean president has said.
Trump and Kim met in Singapore on June 12, marking the first-ever U.S.-North Korea summit.
Moon has invited Kim to visit Seoul before the year's end.
Many believe he may invite Trump to visit Seoul for a three-way summit with Kim as he has repeatedly stressed the need to formally end the Korean War before the year's end, a move that requires the U.S' participation.
The two Koreas technically remain at war as the 1950-53 war ended only with an armistice, not a peace treaty.
Moon insists a formal declaration of an end to the war would simply be a declaration that has nothing to do with the South Korea-U.S. alliance or U.S. troops stationed in South Korea.
Still, it may provide some security assurance to the North, which is exactly what the communist state is seeking in exchange for giving up its nuclear ambition, Moon said.
Following their bilateral summit, Moon and Trump are also expected to sign a revision to the South Korea-U.S. free trade agreement.
The South Korean president will deliver a keynote speech at the U.N. General Assembly on Wednesday. He is also scheduled to hold bilateral summits with his counterparts from Chile and Spain on the sidelines of the U.N. meeting.
He will return home on Thursday.