But some on the left argue that the administration is using China to scare lawmakers and exaggerating the competition. “I just don’t buy this China boogeyman stuff,” said Jared Bernstein, a senior fellow at the Center on Budget and Policy Priorities and a former economic adviser to Vice President Joseph R. Biden Jr. “If anything, I see China as being more inward looking, devoting less resources to mercantile-type trade and more to internal investment, consumption and developing human capital.”

Supporters of the trade pact hope to hold a new vote this week on the part of the trade package rejected by the House on Friday but will need to secure 90 more votes. Representative Paul D. Ryan, the Wisconsin Republican who has worked with the White House to secure trade negotiating authority, said on Sunday that if Mr. Obama wanted to avoid being a “very lame-duck president,” he would have to win over members of his own party.

“I think that this can be salvaged because I think people are going to realize just how big the consequences are for American leadership,” Mr. Ryan said on “Fox News Sunday.”

But there was little indication over the weekend that many minds had been changed on the political left. “We need to regroup and come up with a trade policy which demands that corporate America start investing in this country rather than in countries all over the world,” Senator Bernie Sanders of Vermont, a candidate for the Democratic presidential nomination, said on “Face the Nation” on CBS.

The Pacific trade pact was meant to be one of three major foreign policy achievements Mr. Obama wanted to secure before leaving office; all three are on the line this summer. He faces a June 30 deadline to seal an agreement with Iran to scale back its nuclear program and he hopes to follow through on his reconciliation with Cuba by formally restoring diplomatic relations.

Those three initiatives take on even more significance for Mr. Obama as he confronts the reality that he is likely to turn over the White House to his successor without having defeated the Islamic State in Syria and Iraq or resolved the conflict with Russia over Ukraine. He still plans to leave office as the president who ended the American war in Afghanistan, but the security situation there remains fluid.

Amid all those wartime issues, the Asia initiative was to be the long-term investment that would pay off years later. The White House has deployed more military forces to the region and made it a focus. But the trade pact was to be the most tangible element of the policy.