By Jackie Calmes for The New York Times

WASHINGTON — His successor, whether Democrat or Republican, opposes it, as does most of his party. Delegates at the Democratic National Convention waved signs saying “T.P.P.” slashed by a bold line, while the Republican Party platform opposed any vote on it in Congress this year.

Yet President Obama is readying one final push for approval of the Trans-Pacific Partnership, the largest regional trade agreement ever, between the United States and 11 other Pacific Rim nations. And though the odds may be long, a presidency defined by partisan stalemate may yet secure one last legacy — only because of Mr. Obama’s delicate alliance with the Republicans who control Congress.

“Both parties have candidates who have very strong rhetoric against trade,” said Representative Kevin Brady, Republican of Texas and chairman of the House Ways and Means Committee, which is responsible for trade. “Nonetheless, we can’t grow America’s economy unless we’re not merely buying American but selling American all throughout the globe.”

Still, he added, timing a vote “is absolutely dependent on support for the agreement.”

Although the administration’s push will begin in September, no vote on the accord will occur before the election. Just as the White House and congressional Republican leaders mostly agree on the economic benefits of trade, they have parallel political interests in delaying debate.

Republicans do not want to provoke attacks from their presidential nominee, Donald J. Trump, who called the trade accord “a rape of our country,” or hurt other Republican candidates. Mr. Obama does not want to make trouble for the Democratic nominee, Hillary Clinton, who has struggled to persuade voters of her sincerity in switching from support of the pact to opposition. This month, during an economic address in Michigan, she declared, “I oppose it now, I’ll oppose it after the election and I’ll oppose it as president.”

Yet the administration does not plan to be silent or forfeit hopes for a postelection vote.

Mr. Obama, who advocated the trade accord in a pre-vacation news conference, will rejoin the debate during an early September trip to Asia. Cabinet officials will fan out to promote the agreement, which would end 18,000 tariffs and other nontariff barriers that Japan, Australia and the other nations have against American imports and services, and set new rules for labor and environmental practices.

While administration officials and bipartisan surrogates will counter opponents’ economic arguments, a big focus will be on national security. Mr. Obama has emphasized that the pact would expand American influence in the Asia-Pacific region as a counterweight to China, which is not part of the pact.

Among those who will hit the road will be Secretary of State John F. Kerry; Secretary of Defense Ashton B. Carter; retired Admiral Michael G. Mullen, former chairman of the Joint Chiefs of Staff under Presidents George W. Bush and Obama; Admiral Harry B. Harris Jr., commander of the United States Pacific Command; and William Cohen, a former Republican senator and defense secretary under President Bill Clinton.

Business and agricultural associations have been campaigning through the summer. The Business Roundtable, an association representing the chief executives of some of the largest American companies, has held events in more than 120 congressional districts during lawmakers’ summer recess.

Last week, with families making back-to-school purchases, the lobbying association for footwear companies circulated a report concluding that Americans could save $4 billion on children’s shoes if T.P.P. takes effect and cuts tariffs on imports from Vietnam and elsewhere.

Environmental and labor groups have been active, too, holding “Rock against the T.P.P.” concerts in several cities and flying protest blimps outside lawmakers’ offices.

“Even the most ardent supporters of the bill, which would include us, would say, ‘Please don’t put a bill on the floor if you don’t have the votes,’” said Bill Miller, a vice president at the Business Roundtable. “The parties have been working pretty well to get resolution, but they’re not there yet.”

Talks have begun between administration officials, chiefly Mr. Obama’s trade ambassador, Michael B. Froman, and Republican leaders, including Mr. Brady and Senator Orrin G. Hatch, chairman of the Senate Finance Committee. But essential negotiations over Republican objections are not likely to take place until Republicans decide Congress will act.

Mr. Brady and others cite progress in addressing lawmakers’ concerns about Japanese pork, labor rights in Mexico and financial companies’ data protections. But other issues remain.

Many Republicans and the tobacco industry object that the tobacco companies would be barred from using international trade tribunals to sue countries that restrict smoking. More problematic is the complaint of Republicans, led by Mr. Hatch, and the pharmaceutical industry that the agreement would undercut drug makers’ intellectual property protections on the advanced drugs known as biologics. The issue was the last to be settled among the T.P.P. countries in October; other nations demanded fewer years of protection, to hasten the production of less costly generics.

Mr. Hatch, in a statement, also said he wanted to see written plans from the T.P.P. nations on how they would “abide by their commitments.”

“Those issues have to be addressed in a positive way before we can move forward,” Mr. Brady said, echoing the House speaker, Paul D. Ryan of Wisconsin, and the Senate majority leader, Mitch McConnell of Kentucky. “But the White House really needs to pick up the pace if we’re going to consider it this year.”

Yet satisfying Republicans, who not only run Congress but also provide the bulk of pro-trade votes, risks costing critical Democratic votes.

Senator Ron Wyden of Oregon, senior Democrat on the Senate Finance Committee, said in a statement, “Republicans have made clear they want to gut the public health, pharmaceutical, labor and environmental provisions, which President Obama has said he will not do, and which I strongly oppose.”

The White House cannot afford to lose much support. The template vote is Congress’s narrow approval last year of a “fast track” law that cleared the way for final negotiations on the Pacific pact, by allowing an up-or-down vote without amendments that could unravel the agreement.

In the House, all 28 Democratic supporters remain on board, both sides say. While House Republicans have not counted yet whether they still have at least 190 votes for T.P.P., Mr. Brady said, pro-trade Republicans “are in a good place” if the outstanding issues get resolved.

But the climate has shifted for the traditionally pro-trade party. A new poll from the Pew Research Center found that since May 2015 — just before Mr. Trump began his campaign — the percentage of Republican and Republican-leaning voters with negative views of trade pacts increased 22 percentage points, to 61 percent.

Last week, Senator Patrick Toomey, a Republican fighting for re-election in Pennsylvania, announced he had switched against T.P.P., following Senator Tim Kaine of Virginia, who did so after becoming Mrs. Clinton’s running mate.

Of the 60 senators who supported the fast-track bill last year, at least five have now come out against T.P.P. It will need at least 50 votes, assuming Vice President Joseph R. Biden Jr. is the tiebreaker.

“There is a big gap between the rhetoric of the campaign and even in what you see in the polls,” Mr. Froman said. “It’s going to be hard. But the votes will be there.”