On Tuesday, Wendy Cutler, the acting deputy trade representative, will travel to Japan to complete a bilateral agreement over access to Japan’s automotive and agricultural markets. People with knowledge of those talks say the Obama administration is satisfied with Japan’s offer on auto and auto parts exports, as well as pork, beef and wheat. But Japan’s continued protection of rice farming remains a sticking point.

Mr. Froman is expected to travel to Mexico at the end of the week, with more trips planned for next week.

The closeness of a deal has only amplified the delicacy and complexity of the talks. Democrats and unions are pushing Mr. Obama to hold a hard line on labor rights, especially in Vietnam and Mexico, but if those countries cannot be certified as in compliance with the agreement’s union rules, other parts of the deal cannot be put into effect, said Lori Wallach, a critic of the agreement, with Public Citizen’s Global Trade Watch. That has politicians in countries like Japan worried they will take a hit for signing an agreement that will not be enforced for years.

The United States is pressing Canada to open its agriculture market, but if Washington cuts its own deal with Ottawa, Australia and New Zealand could protest. Peru is questioning whether to accept politically difficult environmental protections for its forests if a promised increase to its exports is held hostage to Mexico’s anti-labor practices.

Australia and New Zealand have protested United States insistence linking the accord’s intellectual property protections with patent rules that could slow the adoption of generic drugs in those countries’ national health programs.

Mr. Obama promised repeatedly that he would deliver the most progressive trade accord in history, and “the time for rhetoric has passed,” Mr. Levin said.

Still, the progress is remarkable for an agreement that started in 2008 with just four countries and the United States interested in only a small part of the talks. Mr. Obama’s promised “pivot” in foreign affairs toward Asia gave the negotiations more momentum. Vietnam entered the talks in 2009, changing the tone as capitalist powers began negotiating rules of commerce with a Communist economy. Japan’s decision to join the talks in 2013 gave the proposed accord enormous economic heft.

“I know there are intense feelings in this country about globalization, but the reality is it’s here to stay. We have to shape it,” said Ron Kirk, Mr. Obama’s first trade representative, who pushed the president into the talks. “The underlying rationale for the United States being involved in these negotiations is still compelling.”