Senate leaders reached a deal Wednesday to revive trade legislation that Democrats had appeared to have scuttled just a day earlier, breathing new life into the centerpiece of President Obama’s economic agenda — though House Democrats vowed a tougher fight still awaits.

The new agreement paves the way for passage of the deal through the Senate, but Mr. Obama’s own troops in the House said their opposition is intensifying, and they blamed the president for mishandling the politics of his top second-term priority.

“If the administration thought it was tough going in the Senate, in the House it will be even harder,” said Rep. Rosa L. DeLauro, Connecticut Democrat and a leading opponent of giving the president special authority to negotiate the trade pact. “I believe they underestimated the depth of feeling.”

Mr. Obama got a taste of defeat Tuesday when all but one Senate Democrat voted to filibuster fast-track trade powers, keeping the entire debate from even reaching the chamber floor. The embarrassing hiccup sent the White House into action, summoning pro-free trade lawmakers to meet with the president.

The administration also scrambled to reassure foreign governments from Japan to Canada that the failed test vote wasn’t fatal to the Trans-Pacific Partnership, which is the trade agreement Mr. Obama is negotiating with 11 other countries. Fast-track trade powers are deemed essential to completing that deal.

In addition to the weighty policy issues, the White House also had to grapple with criticism about Mr. Obama’s bungled personal outreach, as some of his own usual allies accused him as coming off as condescending and even sexist. Liberals from Sen. Sherrod Brown, Ohio Democrat, to NOW President Terry O’Neill said it was sexist for the president to refer to Sen. Elizabeth Warren, Massachusetts Democrat and a vocal opponent of the deal, as “Elizabeth.”

“I think it is sexist,” Ms. O’Neill told The Hill newspaper. “I think the president was trying to build up his own trustworthiness on this issue by convincing us that Senator Warren’s concerns are not to be taken seriously. But he did it in a sexist way.”

Mr. Brown later relented from his criticism, saying he would apologize to Mr. Obama.

On the substance, Democrats signaled they don’t trust Mr. Obama to win a fair trade deal, and said he miscalculated badly by rejecting lawmakers’ calls to include a provision in the TPP to address currency manipulation by other countries.

Rep. David Scott, Georgia Democrat, called it “the Achilles’ heel” of the proposed agreement and said the administration’s refusal to address it was “asinine.”

Rep. Joe Courtney, Connecticut Democrat, said the administration has been downplaying lawmakers’ concerns for years about including an enforceable provision against currency manipulation in the TPP.

“A huge bipartisan collection of members have been sending up smoke signals for years, going back to 2013,” Mr. Courtney said. “To not have an agreement that has enforceable provisions on that issue, it’s almost political malpractice for members of Congress to just sort of look the other way. All of us would have preferred that the administration heeded the early input that all of us were giving on this issue and brought this to the table.”

The administration has argued that restrictions on currency manipulation could boomerang and hurt U.S. monetary policy too. Mr. Courtney, however, rejected that as a “misleading rebuttal.”

Mrs. DeLauro said the administration has “rebuffed” lawmakers’ concerns about currency manipulation, food safety and enforceable labor standards for more than two years, and the problem is now coming back to haunt the president.

“The White House and the administration have only begun to engage in the last few months, really,” she told reporters. “We’ve been rebuffed. Now there is this frenetic sense of ‘let’s try to address those people who are still undecided.’”

The currency debate is what caused Tuesday’s embarrassing hiccup in the Senate as well. Democrats were trying to angle to have a customs enforcement measure, which includes tough currency language, wrapped in with the fast-track bill.

But Republicans and Mr. Obama both insisted that would kill the trade deal. Instead, GOP leaders offered a separate vote this week on the customs bill, giving Democrats a chance for a show vote, but taking the teeth out of it.

That smoothed over the first hurdle, but all sides expect more.

Speaker John A. Boehner, Ohio Republican, who supports the deal, has said the White House needs to provide about 50 votes from the 188 Democratic House lawmakers to help pass the measure. More than 150 House Democrats last year came out in opposition to the deal, and Mrs. DeLauro recited the names of a handful of other Democratic lawmakers who have recently announced their opposition, including Reps. Adam B. Schiff of California and Tammy Duckworth of Illinois.

The so-called “fast-track” authority allows a president to negotiate trade deals with other countries, then submit them for up-or-down votes in Congress without amendments. Analysts say that if Congress were allowed to amend deals, it would make it impossible to negotiate with foreign governments.

But the Senate test vote brought to light the breathtaking spectacle of Democratic lawmakers complaining that the president of their own party was trying to grab too much power.

“We have seen too much power ceded from Congress to the executive branch at the expense of the United States, particularly in foreign matters,” Mr. Scott said. “You can look at the Iran [nuclear] deal, look at a lot of things. Congress has to retain our place. This should be a shared trade deal, not just by the president himself.”

The White House also faced more second-guessing about Mr. Obama’s decision to promote the trade deal last week at the headquarters of Nike, a company that has used “sweatshop” labor overseas — a particular sore point with progressive groups.

The simmering feud between Mr. Obama and Mrs. Warren, an icon of the progressive wing, also showed no signs of abating, with Mr. Earnest tossing more criticism her way, saying her claims that the pact is being negotiated in secret “are just not true.”

“If people do want to raise a substantive objection to pursuing this strategy, then they should stick to the facts,” Mr. Earnest said. We’re interested in a substantive debate. But we don’t have to say things that aren’t true in order to have this debate.”

Mr. Earnest also defended the decision to visit Nike.

“This is the debate that we want to have, because this is the substance of the debate,” he said. “We are well aware that there are people in both parties that have raised concerns about Nike’s previous practices in terms of doing business overseas. The question right now is, ‘What are we going to do about it?’ And what we see from progressives is a lot of complaining about it.”

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