“The Chinese government’s response is to build the free-trade agreements that it can influence,” said Li Daokui, a professor at the School of Economics and Management at Tsinghua University in Beijing. “I would say it was a mistake for the U.S. not to include China. If China had been allowed to join at the beginning, the landscape would be entirely different.”

Mr. Li said that if China had been part of the negotiations at the start, it might not have pushed so hard across the region for its own trade agreements, and “everything would be much simpler.”

Perhaps for Mr. Xi. For Mr. Obama, it might have been even more complex. The domestic politics of this deal are difficult enough without China as a potential signatory, as shown by his failure to prevail in an important vote in the Senate on Tuesday to get the authority he needs to negotiate the trade agreement. (Administration officials insisted it was a temporary and tactical setback.)

Kevin Rudd, the former Australian prime minister, has spent the last year at Harvard and at the Asia Society studying the long-term future between China and the West. He concluded in a recently published report that in both China and the United States there was a rush “to believe that the two countries are now locked into some sort of irreversible and increasingly fractious zero-sum game.” Instead, he found the relationship was still “functioning reasonably effectively,” but noted that Mr. Xi had changed strategy.

“He has ended former paramount leader Deng Xiaoping’s foreign policy orthodoxy over the past 35 years of ‘hide your strength, bide your time, never take the lead,’ in favor of a more vigorous, activist and assertive international policy to advance Chinese interests both in the region and beyond,” he wrote in Foreign Policy.

Shi Yinhong, a professor of international relations at Renmin University, said Chinese leaders were well aware of how important approval of the trade agreement, also known as the TPP, is for Mr. Obama as he tries what he has often called a “rebalancing” toward Asia.

“The Chinese government knows the TPP is a major attempt by the U.S. to win back economic leadership in the region,” Mr. Shi said. “China also knows the Asia Pacific region is such a wide region, so you can have two stages. One is led by the U.S., which is pushing the TPP. The other is dominated by China.”