Shinzo Abe said he wanted to "build trust" and "work together for prosperity and world peace", before leaving for his trip, the BBC reported.
The meeting in New York comes amid concern over the foreign policy direction of Tokyo's biggest ally.
Mr. Trump has said Japan needs to pay more to maintain U.S. troops on its soil.
He also condemned a major trade deal struck by President Obama with Japan and other Pacific Rim countries.
The U.S. and Japan have been key allies since the end of World War Two, when the U.S. helped Japan rebuild its economy.
Mr. Abe is stopping in New York on his way to an Asia-Pacific trade summit in Peru.
But details of Thursday's meeting are unclear, with a Japanese official saying exactly where it will happen has not been firmed up.
"There has been a lot of confusion," a Japanese official told Reuters news agency.
"We want to safeguard our alliance with the United States during the transition," said a senior Japanese foreign ministry official, Tetsuya Otsuru, when the meeting was announced.
Read alsoEuropean leaders, NATO caution Trump on warming relations with Russia - WPThe BBC's Paul Adams in Washington says Mr. Trump has toned down some of his more strident rhetoric since the election, but that hasn't stopped observers wondering whether the fundamentals of post-war U.S. policy on Asia can survive.
High-level talks are rarely held in such an informal context and Tokyo is keen to minimize uncertainty during the long handover of power.
