Putin and Trump first met at a G20 summit in Hamburg in July when they discussed allegations of Russian meddling in the U.S. election last year but agreed to focus on better ties rather than litigating the past.
Trump leaves behind a spate of scandals behind, including the indictment of his former campaign chairman Paul Manafort.
The final leg of Trump's trip will be spent in the Philippines, where he will attend the U.S.-ASEAN and East Asia summits and meet for the first time with Philippine President Rodrigo Duterte.
Trump made the comments as relations between Moscow and Washington have soured since allegations of Russian meddling in the November general election began to surface.
Trump in August grudgingly signed off on new sanctions against Russia, a move Moscow said ended hopes for better ties. Putin ordered Washington to cut its embassy and consular staff in Russia by more than half.
Tensions have also flared over the conflict in Syria.
If the Trump-Putin meeting comes about, it would come as investigations in Washington over alleged Russian meddling in the presidential election and possible collusion by the Trump campaign yielded its first indictments.
U.S. special counsel Robert Mueller's office this week unveiled charges against former Trump campaign manager Paul Manafort, Manafort associate Richard Gates and campaign foreign policy adviser George Papadopoulos. Manafort and Gates pleaded not guilty while Papadapoulos pleaded guilty.
The Wall Street Journal also has reported that U.S. authorities have enough evidence to charge six members of the Russian government in the hacking of Democratic National Committee computers during the 2016 campaign.
