In remarks during a bilateral meeting on Saturday between Mr. Obama and President Xi Jinping of China, Mr. Xi said the relationship between their countries was at “a hinge moment” and added that China would work with others to ensure a successful summit meeting.
“I hope the two sides will work together to focus on cooperation, manage our differences and make sure there is a smooth transition in the relationship, and that it will continue to grow going forward,” he said.
Mr. Obama will find that his counterparts are already getting pulled deeper into China’s economic riptide because of the pact’s demise.
Australia said on Wednesday that it wanted to push ahead with a Chinese-led trade pact that would cover Asian nations from Japan to India but exclude the United States. Peru has opened talks with Beijing to join the agreement as well. Even American business leaders are positioning themselves for the potential opportunities in Asia.
“Two-thirds of what we do there ends up in another country,” said John G. Rice, General Electric’s vice chairman for international operations. “So if they’re going to lower tariffs and trade barriers within that region, we’ll find ways to do more there.”
For the United States, such trade ties have geopolitical undertones.
Much of Asia has for decades quietly accepted American security guarantees while also running large trade surpluses with the United States, turning them into prosperous manufacturing powerhouses. But China is now the largest trading partner for most of the region, while at the same time making territorial claims against many of its neighbors.
The neighbors fear they could soon face a stark choice among money, pride and place: Accede to China’s security demands, or lose access to China’s vast market.
“The long-term question is whether America pulls back from Asia and makes it easier for China to force countries in the region to make a choice between China and the United States,” said Richard Bush, a senior fellow at the Brookings Institution.
When it comes to Asia, the shift in American policy is likely to be among the most pronounced in the new administration.
Born in Hawaii, in the middle of the Pacific Ocean, and raised partly in Indonesia, Mr. Obama has, since his earliest days as president, pushed Asia as his nation’s best opportunity for growth. Other than efforts to end two wars begun by his predecessor, Mr. Obama’s principal foreign policy goal was to shift the nation’s attention and military resources away from an aging Europe and an unproductive rivalry with Russia and toward the fast-growing Asia-Pacific region.
It is a goal he partly shared with President George W. Bush, whose administration began the negotiations for the Trans-Pacific Partnership. Mr. Bush, too, saw the pact as a bulwark against China’s rising strategic ambitions.
Mr. Obama has spent much of the last 10 days putting a brave face on the crushing blow that Mr. Trump’s surprising electoral victory represented to his legacy.
Hours after the results became clear, he told the country and his own weeping staff that “we’re all actually on one team.” On Monday, he suggested Republicans would find replacing his health care law difficult. And in visits on Tuesday and Thursday to Greece and Germany, he told panicked Europeans that Mr. Trump would preserve the NATO alliance.
But the meetings this weekend, at the Asia-Pacific trade summit, will very likely have a more funereal feel.
More than on any other issue, Mr. Obama and Mr. Trump have diametrically opposing views on Asian trade. And Mr. Trump has won. Even White House aides have admitted to their boss’s bitter defeat.
“We’re cleareyed about the current situation, but we believe what we believe about the value of trade and the importance of the Asia-Pacific region to the United States,” said Benjamin J. Rhodes, a deputy national security adviser.
The allegiances are starting to shift after Mr. Trump’s election.
Just three days before Mr. Obama’s arrival here, Peru’s foreign minister, Eduardo Ferreyros, said the country still hoped the Pacific pact would someday become a reality. But given the changing dynamics, his government also opened talks this autumn with Beijing to join the rival, Chinese-led trade pact, the Regional Comprehensive Economic Partnership.
“Since Mr. Trump is not so interested in requiring economic integration and trade liberalization, why not have other countries follow this free-trade proposal?” asked Song Guoyou, a longtime trade specialist who is the deputy director of the Center for American Studies at Fudan University in Shanghai.
Since the election, Australia’s government has also called for rapid progress in concluding that rival trade pact. Even Japan, despite facing territorial demands from China and close, but peaceful, confrontations between the two countries’ military jets and coast guard vessels, is paying more attention to China’s vision for global trade.
Australia and Japan have been bargaining for years with China on the deal. But they wanted it as a complement to the Trans-Pacific Partnership, to balance their economic relationship with the United States instead of replacing it with ties to China.
“If T.P.P. doesn’t move forward, there’s no doubt that the focus will shift” to the China-led deal, Prime Minister Shinzo Abe of Japan told his country’s Parliament on Tuesday. Mr. Abe met with Mr. Trump on Thursday.
Since 2011, trade negotiators from China, Japan, Australia, India and 12 other Asian nations have been meeting several times a year to stitch together the Regional Comprehensive Economic Partnership. And with Mr. Trump’s victory, those efforts are almost certain to accelerate. The next round of talks is to be held in Indonesia early next month.
Trade officials across Asia met to negotiate details in Cebu, the Philippines, the week before Mr. Trump won the election. Almost no one noticed outside of Cebu. The next meeting, scheduled for early December, could attract far more attention, including some at this weekend’s Asia-Pacific Economic Cooperation summit meeting in Lima.
“Everyone is waiting to see what will happen at the APEC summit — there will be a lot of discussions on the sidelines on how to deal with Trump,” said Kishore Mahbubani, the dean of the Lee Kuan Yew School of Public Policy at the National University of Singapore, said of this weekend’s meeting. “The biggest winner if you walk away from T.P.P. is China. There’s no doubt about that.”