White House officials had said that a meeting by Mr. Trump and Mr. Putin — who last held face-to-face talks in Hamburg, Germany, in July — was a possibility on the sideline of the upcoming Asia-Pacific Economic Cooperation summit in Da Nang, Vietnam.

“We want Putin’s help on North Korea,” Mr. Trump said.

The White House also signaled that Mr. Trump could decide on this trip to designate North Korea a state sponsor of terrorism, a largely symbolic move since it is already among the world’s most heavily sanctioned countries. Still, the gesture would reinforce the administration’s efforts to cast the North as a global pariah.

Rex W. Tillerson, the secretary of state, and other administration officials are looking at the prospect “very closely,” a senior White House official told reporters in Tokyo on Sunday, and a decision will come “very soon.”

The president used his speech on Sunday to call for building a “free and open Indo-Pacific” region, a new approach to Asia that is likely to be seen by China as a challenge. The idea, first proposed by the Japanese and adopted in recent days by Mr. Tillerson, envisions the United States strengthening ties with three other democracies in the region — Japan, Australia and India — to contain a rising China.

“We will seek new opportunities for cooperation and commerce, and we will partner with friends and allies to pursue a free and open Indo-Pacific region,” Mr. Trump said. “We will seek free fair and reciprocal trade.”

Mr. Trump’s trip to the continent will be the longest by an American president in more than 25 years, with additional stops in South Korea, China and the Philippines. Ahead of what his advisers called a grueling schedule of meetings and summits, the president got a chance to relax by playing golf on Sunday afternoon with Prime Minister Shinzo Abe of Japan at Kasumigaseki Country Club.

The outing was reciprocation for rounds that Mr. Trump hosted in Jupiter, Fla. and nearby West Palm Beach in February for Mr. Abe and Ernie Els, once the world’s top golfer. For Sunday’s round, Mr. Abe invited Hideki Matsuyama, a Japanese golfer ranked fourth in the world. Before the game, the Japanese prime minister presented Mr. Trump with white caps in the style of the president’s trademark red “Make America Great Again” trucker hats; Mr. Abe’s were emblazoned in gold: “Donald and Shinzo Make Alliance Even Greater.”